Quantum limits in optical interferometry
aa r X i v : . [ qu a n t - ph ] A ug , Quantum limits in optical interferometry
R. Demkowicz-Dobrzański, M. Jarzyna, and J. Kołodyński
Faculty of Physics,University of Warsaw,ul. Hoża 69, PL-00-681 Warszawa,Poland
Non-classical states of light find applications in enhancing the performance of opticalinterferometric experiments, with notable example of gravitational wave-detectors. Still,the presence of decoherence hinders significantly the performance of quantum-enhancedprotocols. In this review, we summarize the developments of quantum metrology withparticular focus on optical interferometry and derive fundamental bounds on achiev-able quantum-enhanced precision in optical interferometry taking into account the mostrelevant decoherence processes including: phase diffusion, losses and imperfect interfer-ometric visibility. We introduce all the necessary tools of quantum optics as well asquantum estimation theory required to derive the bounds. We also discuss the practi-cal attainability of the bounds derived and stress in particular that the techniques ofquantum-enhanced interferometry which are being implemented in modern gravitationalwave detectors are close to the optimal ones.
CONTENTS
I. Introduction 1II. Quantum states of light 4A. Mode description 4B. Gaussian states 41. Coherent states 52. Single-mode squeezed states 53. Two-mode squeezed states 64. General two-mode Gaussian states 6C. Definite photon number states 7D. Particle description 7E. Mode vs particle entanglement 8III. Mach-Zehnder interferometry 8A. Phase-sensing uncertainty 9B. Coherent-state interferometry 10C. Fock state interferometry 10D. Coherent + squeezed-vacuum interferometry 10E. Definite photon-number state interferometry 11F. Other interferometers 12IV. Estimation Theory 13A. Classical parameter estimation 141. Fisher Information approach 142. Bayesian approach 153. Example: Transmission coefficient estimation 16B. Quantum parameter estimation 171. Quantum Fisher Information approach 172. Bayesian approach 19V. Quantum limits in decoherence-free interferometry 19A. Quantum Fisher Information approach 20B. Bayesian approach 21C. Indefinite photon-number states 221. Role of the reference beam 222. Optimal indefinite photon number strategies 233. Gaussian states 23D. Role of entanglement 24E. Multi-pass protocols 25VI. Quantum limits in realistic interferometry 25A. Decoherence models 26 1. Phase diffusion 272. Photonic losses 273. Imperfect visibility 28B. Bounds in the QFI approach 281. Imperfect visibility 292. Photonic losses 303. Phase diffusion 32C. Bayesian approach 331. Photonic losses 332. Phase diffusion 34D. Practical schemes saturating the bounds 341. Bounds for indefinite photon number states 342. Coherent + squeezed vacuum strategy 35VII. Conclusions 36References 37
I. INTRODUCTION
Without much exaggeration one may say that optics isbasically the science of light interference. Light interfer-ence effects were behind the final acceptance of classicalwave optics and abandoning the Newtonian corpusculartheory of light. Conceptual insight into the process ofinterfering light waves prompted development of a num-ber of measurement techniques involving well controlledinterference effects and gave birth to the field of opticalinterferometry (Hariharan, 2003). At the fundamentallevel, classical interferometry is all about observing lightintensity variations (intensity fringes) resulting from achange in relative phases between two (or more) overlap-ping light waves, e.g. when a single light beam is splitinto a number of paths with tunable optical path-lengthdifferences and made to interfere on the screen. Thenumber of applications is stunning and ranges from ba-sic length measurements via spectroscopic interferomet-ric techniques to the most spectacular examples involv-ing stellar interferometry and gravitational-wave detec-tors (Pitkin et al. , 2011).Coherent properties of light as well as the degree ofoverlap between the interfering beams determine the vis-ibility of the observed intensity fringes and are crucialfor the quality of any interferometric measurement. Still,when asking for fundamental limitations on precision ofestimating e.g. a phase difference between the arms of aMach-Zehnder interferometer, there is no particular an-swer within a purely classical theory, where both the lightitself as well as the detection process are treated classi-cally. In classical theory intensity of light can in principlebe measured with arbitrary precision and as such allowsto detect in principle arbitrary small phase shifts in aninterferometric experiment.This, however, is no longer true when semi-classicaltheory is considered in which the light is still treatedclassically but the detection process is quantized, so thatinstead of a continuous intensity parameter the numberof energy quanta (photons) absorbed is being measured.The absorption process within the semi-classical theoryhas a stochastic character and the number of photonsdetected obeys Poissonian or super-Poissonian statistics(Fox, 2006). If light intensity fluctuations can be ne-glected, the number of photons detected, N , follows thePoissonian statistics with photon number standard devia-tion ∆ N = p h N i , where h N i denotes the mean numberof photons detected. This implies that the determina-tion of the relative phase difference ϕ between the armsof the interferometer, based on the number of photonsdetected at the output ports, will be affected by the rel-ative uncertainty ∆ ϕ ∝ ∆ N/ h N i = 1 / p h N i referredto as the shot noise . The shot noise plays a fundamen-tal role in the semi-classical theory and in many casesit is indeed the factor limiting achievable interferometricsensitivities. In modern gravitational wave detectors, inparticular, the shot noise is the dominant noise term inthe noise spectral density for frequencies above few hun-dred Hz (LIGO Collaboration, 2011, 2013; Pitkin et al. ,2011).Yet, the shot noise should not be regarded as a fun-damental bound whenever non-classical states of lightare considered—we review the essential aspects of non-classical light relevant from the point of view of inter-ferometry in Sec. II. The sub-Poissonian statistics char-acteristic for the so-called squeezed states of light mayoffer a precision enhancement in interferometric scenar-ios by reducing the photon number fluctuations at theoutput ports. First proposals demonstrated that sendingcoherent light together with the squeezed vacuum stateinto the two separate input ports of a Mach-Zehnder in-terferometer offers estimation precision beating the shotnoise and attaining the / h N i / scaling of the phaseestimation precision (Caves, 1981), while considerationof more general two-mode squeezed states showed thateven / h N i scaling is possible (Bondurant and Shapiro, 1984; Yurke et al. , 1986). A number of papers followed,studying in more detail the phase-measurement probabil-ity distribution and proposing various strategies leadingto / h N i precision scaling (Braunstein, 1992; Dowling,1998; Holland and Burnett, 1993; Sanders and Milburn,1995). Similar observations have been made in the con-text of precision spectroscopy, where spin-squeezed stateshave been shown to offer a /N scaling of atomic tran-sition frequency estimation precision, where N denotesthe number of atoms employed (Wineland et al. , 1994,1992). Sec. III provides a detailed framework for deriv-ing the above results. However, already at this point abasic intuition should be conveyed that one cannot gobeyond the shot noise limit whenever an interferometricexperiment may be regarded in a spirit that each pho-ton interferes only with itself. In fact, the only possibil-ity of surpassing this bound is to use light sources thatexhibit correlations in between the constituent photons,e.g. squeezed light, so that the interference process maybenefit from the properties of inter-photonic entangle-ment.These early results provided a great physical insightinto the possibilities of quantum enhanced interferometryand the class of states that might be of practical interestfor this purpose. The papers lacked generality, however,by considering specific measurement-estimation strate-gies, in which the error in the estimated phase was relatedvia a simple error-propagation formula to the variance ofsome experimentally accessible observable, e.g. photonnumber difference at the two output ports of the inter-ferometer, or by studying the width of the peaks in theshape of the phase-measurement probability distribution.Given a particular state of light fed into the interferom-eter, it is a priori not clear what is the best measurementand estimation strategy yielding the optimal estimationprecision. Luckily, the tools designed to answer thesekinds of questions had already been present in the lit-erature under the name of quantum estimation theory (Helstrom, 1976; Holevo, 1982). The Quantum FisherInformation (QFI) as well as the cost of Bayesian infer-ence provide a systematic way to quantify the ultimatelimits on performance of phase-estimation strategies fora given quantum state, which are already optimized overall theoretically admissible quantum measurements andestimators. The concept of the QFI and the Bayesianapproach to quantum estimation are reviewed in Sec. IV.As a side remark, we should note, that by treating thephase as an evolution parameter to be estimated and sep-arating explicitly the measurement operators from theestimator function, quantum estimation theory circum-vents some of the mathematical difficulties that arise ifone insists on the standard approach to quantum mea-surements and attempts to define the quantum phase op-erator representing the phase observable being measured(Barnett and Pegg, 1992; Lynch, 1995; Noh et al. , 1992;Summy and Pegg, 1990).The growth of popularity of the QFI in the field ofquantum metrology was triggered by the seminal paper ofBraunstein and Caves (1994) advocating the use of QFIas a natural measure of distance in the space of quantumstates. The QFI allows to pin down the optimal probestates that are the most sensitive to small variations ofthe estimated parameter by establishing the fundamentalbound on the corresponding parameter sensitivity validfor arbitrary measurements and estimators. Followingthese lines of reasoning the /N bound, referred to asthe Heisenberg limit , on the phase estimation precisionusing N -photon states has been claimed fundamental andthe NOON states were formally proven to saturate it(Bollinger et al. , 1996). Due to close mathematical analo-gies between optical and atomic interferometry (Bollinger et al. , 1996; Lee et al. , 2002) similar bounds hold for theproblem of atomic transition-frequency estimation andmore generally for any arbitrary unitary parameter esti-mation problem, i.e. the one in which an N -particle stateevolves under a unitary U ⊗ Nϕ , U ϕ = exp( − i ˆ Hϕ ) , with ˆ H being a general single-particle evolution generator and ϕ the parameter to be estimated (Giovannetti et al. , 2004,2006).A complementary framework allowing to determine thefundamental bounds in interferometry is the Bayesian ap-proach, in which one assumes the estimated parameter tobe a random variable itself and explicitly defines its prior distribution to account for the initial knowledge about ϕ before performing the estimation. In the case of interfer-ometry the typical choice is the flat prior p ( ϕ ) = 1 / π which reflects the initial ignorance of the phase. Thesearch for the optimal estimation strategies within theBayesian approach is possible thanks to the general the-orem on the optimality of the covariant measurements in estimation problems satisfying certain group symme-try (Holevo, 1982). In the case of interferometry, a flatprior guarantees the phase-shift, U(1), symmetry andas a result the optimal measurement operators can begiven explicitly and they coincide with the eigenstates ofthe Pegg-Barnett “phase operator” (Barnett and Pegg,1992). This makes it possible to optimize the strategyover the input states and for simple cost functions allowsto find the optimal probe states (Berry and Wiseman,2000; Bužek et al. , 1999; Luis and Perina, 1996). In par-ticular, for the ( δϕ/ cost function which approx-imates the variance for small phase deviations δϕ , thecorresponding minimal estimation uncertainty has beenfound to read ∆ ϕ ≈ π/N for large N providing again aproof of the possibility of achieving the Heisenberg scal-ing, yet with an additional π coefficient. It should benoted that the optimal states in the above approach thathave been found independently in (Berry and Wiseman,2000; Luis and Perina, 1996; Summy and Pegg, 1990)have completely different structure to the NOON stateswhich are optimal when QFI is considered as the figureof merit. This is not that surprising taking into account that the NOON states suffer from the π/N ambiguityin retrieving the estimated phase, and hence are designedonly to work in the local estimation approach when phasefluctuations can be considered small. Derivations of theHeisenberg bounds for phase interferometry using boththe QFI and Bayesian approaches are reviewed in Sec. V.We also discuss the problem of deriving the bounds forstates with indefinite photon number in which case re-placing N in the derived bounds with the mean numberof photons h N i is not always legitimate, so that in somecases the “naive” Heisenberg bound / h N i may in prin-ciple be beaten (Anisimov et al. , 2010; Giovannetti andMaccone, 2012; Hofmann, 2009).Further progress in theoretical quantum metrologystemmed from the need to incorporate realistic deco-herence processes in the analysis of the optimal esti-mation strategies. While deteriorating effects of noiseon precision in quantum-enhanced metrological proto-cols have been realized by many authors working in thefield (Caves, 1981; Datta et al. , 2011; Gilbert et al. ,2008; Huelga et al. , 1997; Huver et al. , 2008; Rubin andKaushik, 2007; Sarovar and Milburn, 2006; Shaji andCaves, 2007; Xiao et al. , 1987), it has long remained anopen question to what extent decoherence effects maybe circumvented by employing either more sophisticatedstates of light or more advanced measurements strategiesincluding e.g. adaptive techniques.With respect to the most relevant decoherence pro-cess in optical applications, i.e. the photonic losses,strong numerical evidence based on the QFI (Demkowicz-Dobrzanski et al. , 2009; Dorner et al. , 2009) indicatedthat in the asymptotic limit of large number of pho-tons the precision of the optimal quantum protocolsapproaches const / √ N , and hence the gain over classi-cal strategies is bound to a constant factor. This facthas been first rigorously proven within the Bayesian ap-proach (Kołodyński and Demkowicz-Dobrzański, 2010)and then independently using the QFI (Knysh et al. ,2011). Both approaches yielded the same fundamentalbound on precision in the lossy optical interferometry: ∆ ϕ ≥ p (1 − η ) / ( ηN ) , where η is the overall power trans-mission of an interferometric setup. This bound is alsovalid after replacing N with h N i when dealing with statesof indefinite photon number, and moreover can be easilysaturated using the most popular scheme involving a co-herent and a squeezed-vacuum state impinged onto twoinput ports of the Mach-Zehnder interferometer (Caves,1981). This fact also implies that the presently imple-mented quantum enhanced schemes in gravitational wavedetection, based on interfering the squeezed vacuum withcoherent light, operate close to the fundamental bound(Demkowicz-Dobrzański et al. , 2013), i.e. they make al-most optimal use of non-classical features of light for en-hanced sensing given light power and loss levels presentin the setup. Based on the mathematical analysis of thegeometry of quantum channels (Fujiwara and Imai, 2008;Matsumoto, 2010) general frameworks have been devel-oped allowing to find fundamental bounds on quantumprecision enhancement for general decoherence models(Demkowicz-Dobrzański et al. , 2012; Escher et al. , 2011).These tools allow to investigate optimality of estimationstrategies for basically any decoherence model and typ-ically provide the maximum allowable constant factorimprovements forbidding better than / √ N asymptoticscaling of precision. Detailed presentation of the abovementioned results is given in Sec. VI.Other approaches to derivation of fundamental metro-logical bounds have been advocated recently. Makinguse of the calculus of variations it was shown in (Knysh et al. , 2014) how to obtain exact formulas for the achiev-able asymptotic precision for some decoherence models,while in (Alipour, 2014; Alipour et al. , 2014) a variantsof QFI have been considered in order to obtain easier tocalculate, yet weaker, bounds on precision. While de-tailed discussion of these approaches is beyond the scopeof the present review, in Sec. VI.B.3 we make use of theresult from (Knysh et al. , 2014) to benchmark the preci-sion bounds derived in the case of phase diffusion noisemodel.The paper concludes with Sec. VII with a summaryand an outlook on challenging problems in the theory ofquantum enhanced metrology. II. QUANTUM STATES OF LIGHT
The advent of the laser, light-squeezing and single-photon light sources triggered developments in interfer-ometry that could benefit from the non-classical featuresof light (Buzek and Knight, 1995; Chekhova, 2011; Torres et al. , 2011). In this section, we focus on the quantum-light description of relevance to quantum optical inter-ferometry. We discuss the mode description of light andthe most commonly used states in quantum optics—theGaussian states. In the end, we consider states of defi-nite photon-number and study their particle-description,in particular, investigating their relevant entanglementproperties.
A. Mode description
Classically, electromagnetic field can be divided intoorthogonal modes distinguished by their characteristicspatial, temporal and polarization properties. This fea-ture survives in the quantum description of light, whereformally we may associate a separate quantum subsys-tem with each of these modes. Each subsystem is de-scribed by its own Hilbert space and, because photonsare bosons, can be occupied by an arbitrary number ofparticles (Mandel and Wolf, 1995; Walls and Milburn,1995). The most general M -mode state of light may be then written as: ρ = X n , n ′ ρ n , n ′ | n ih n ′ | , Tr ( ρ ) = 1 , ρ ≥ (1)with n = { n , . . . , n M } and | n i = | n i ⊗ · · · ⊗ | n M i rep-resenting a Fock state with exactly n i photons occupyingthe i -th mode. States | n i i may be further expressed interms of the respective creation and annihilation opera-tors ˆ a † i , ˆ a i obeying [ˆ a i , ˆ a † j ] = δ ij : | n i i = ˆ a † ni √ n ! | i , ˆ a | n i i = √ n i | n i − i , ˆ a † i | n i i = √ n i + 1 | n i +1 i , (2)where | i is the vacuum state with no photons at all.In the context of optical interferometry, modes are typ-ically taken to be distinguishable by their spatial sepa-ration, corresponding to different arms of an interferom-eter, whereas the various optical devices such as mir-rors, beam-splitters or phase-delay elements transformthe state on its way through the interferometer. Eventu-ally, photon numbers are detected in the output modesallowing to infer the value of the relative phase differencebetween the arms of the interferometer.In many applications, the above standard state repre-sentation may not be convenient and phase-space descrip-tion is used instead—in particular, the Wigner functionrepresentation (Schleich, 2001; Wigner, 1932). Adoptingthe convention in which the quadrature operators read ˆ x i = ˆ a † i + ˆ a i and ˆ p i = i(ˆ a † i − ˆ a i ) , the Wigner functionmay be regarded as a quasi-probability distribution onthe quadrature phase space: W ( x , p ) = 1(2 π ) M ˆ d M x ′ d M p ′ Tr (cid:16) ρ e i[ p ′ (ˆ x − x ) − x ′ (ˆ p − p )] (cid:17) , (3)where x = { x , . . . , x M } , ˆ x = { ˆ x , . . . , ˆ x M } and similarlyfor p and ˆ p . As a consequence, the Wigner function isreal, integrates to over the whole phase space and itsmarginals yield the correct probability densities of eachof the phase space variables. Yet, since it may take nega-tive values it cannot be regarded as a proper probabilitydistribution. Most importantly, it may be reconstructedfrom experimental data either by tomographic methods(Schleich, 2001) or by direct probing of the phase space(Banaszek et al. , 1999), and hence is an extremely use-ful representation both for theoretical and experimentalpurposes. B. Gaussian states
From the practical point of view, the most interestingclass of states are the Gaussian states (D’Ariano et al. ,1995; Olivares and Paris, 2007; Pinel et al. , 2012, 2013).The great advantage of using them is that they are rela-tively easy to produce in the laboratory with the help ofstandard laser-sources and non-linear optical elements,which allow to introduce non-classical features such assqueezing or entanglement. Gaussian states have foundnumerous application in various fields of quantum infor-mation processing (Adesso, 2006) and are also extensivelyemployed in quantum metrological protocols.Gaussian states of M modes are fully characterizedby their first and second quadrature moments and aremost conveniently represented using the Wigner functionwhich is then just a multidimensional Gaussian distribu-tion W ( z ) = 1(2 π ) M √ det σ e − ( z −h ˆ z i ) T σ − ( z −h ˆ z i ) , (4)where for a more compact notation we have introduced:the phase space variable z = { x , p , . . . , x M , p M } ,the vector containing mean quadrature values h ˆ z i = {h ˆ x i , h ˆ p i , . . . , h ˆ x M i , h ˆ p M i} , h ˆ z i i = Tr (ˆ z i ρ ) = ´ d M z W ( z ) z i , and the M dimensional covariance ma-trix σ : σ i,j = 12 h ˆ z i ˆ z j + ˆ z j ˆ z i i − h ˆ z i ih ˆ z j i . (5)Gaussian states remain Gaussian under arbitrary evo-lution involving Hamiltonians at most quadratic in thequadrature operators, what includes all passive devicessuch as beam-splitters and phase-shifters as well assingle- and multi-mode squeezing operations. Below wefocus on a few classes of Gaussian states highly relevantto interferometry.
1. Coherent states
Coherent states are the Gaussian states with identitycovariance matrix σ =
1, so that the uncertainties areequal for all quadratures saturating the Heisenberg un-certainty relations ∆ x i ∆ p i = 1 and there are no cor-relations between the modes. Mean values of quadra-tures may be arbitrary and correspond to the coherentstate complex amplitude α = ( h ˆ x i + i h ˆ p i ) / . These arethe states produced by any phase-stabilized laser, whatmakes them almost a fundamental tool in the theoret-ical description of many quantum optical experiments.Moreover, coherent sates have properties that resemblefeatures of classical light, and thus enable to establish abridge between the quantum and classical descriptions oflight.In the standard representation, an M -mode coherentstate | α i = | α i⊗· · ·⊗| α M i is a tensor product of single-mode coherent states, whereas a single-mode coherentstate is an eigenstate of the respective annihilation oper-ator: ˆ a | α i = α | α i , (6) where α = | α | e iθ and | α | , θ are respectively the ampli-tude and the phase of a coherent state. Equivalently, wemay write | α i = ˆ D ( α ) | i , (7)where ˆ D ( α ) = e α ˆ a † − α ∗ ˆ a is the so-called displacement op-erator or write the coherent state explicitly as a super-position of consecutive Fock states: | α i = e −| α | / ∞ X n =0 α n √ n ! | n i . (8)From the formula above, it is clear that coherent states donot have a definite photon number and if a photon num-ber n is measured its distribution follows the Poissonianstatistics P ( n ) = e −| α | | α | n n ! with average h n i = | α | andstandard deviation ∆ n = | α | . Thus, the relative uncer-tainty ∆ n/ h n i in the measured photon number scales like / p h n i and hence in the classical limit of large h n i thebeam power may be determined up to arbitrary precision.Moreover, the evolution of the coherent state amplitudeis identical to the evolution of a classical-wave ampli-tude. In particular, an optical phase delay ϕ transformsthe state | α i into | α e i ϕ i . These facts justify a commonjargon of calling coherent states the classical states oflight, even though for relatively small amplitudes differ-ent coherent states may be hard to distinguish due totheir non-orthogonality |h α | β i| = e | α − β | . More gener-ally, we call ρ cl a classical state of light if and only if itcan be written as a mixture of coherent states: ρ cl = ˆ d M α P ( α ) | α ih α | (9)with P ( α ) ≥ , which is equivalent to the statementthat ρ cl admits a non-negative Glauber P -representation(Glauber, 1963; Walls and Milburn, 1995). Classicalstates are often used as a benchmark to test the degree ofpossible quantum enhancement which may be obtainedby using more general states outside this class.
2. Single-mode squeezed states
Heisenberg uncertainty principle imposes that ∆ x ∆ p ≥ for all possible quantum states. Single-mode states that saturate this inequality are calledthe single-mode squeezed states (Walls and Milburn,1995). As mentioned above, coherent states fall intosuch a category serving as a special example for which ∆ x = ∆ p = 1 . Yet, as for general squeezed states ∆ x = ∆ p , the noise in one of the quadratures can bemade smaller than in the other. Formally, a single-modesqueezed state may always be expressed as | α, r i = ˆ D ( α ) ˆ S ( r ) | i , (10) (a) (b) FIG. 1 Phase-space diagrams denoting uncertainties in dif-ferent quadratures for momentum squeezed states (a) and forposition squeezed states (b). Dashed circles denote corre-sponding uncertainties for coherent states ∆ x = ∆ p . where ˆ S ( r ) = exp( r ∗ ˆ a − r ˆ a † ) is the squeezing opera-tor , r = | r | e i θ is a complex number and | r | and θ are the squeezing factor and the squeezing angle respectively. Infact, any pure Gaussian one-mode state may be writtenin the above form. For θ = 0 , uncertainties in the quadra-tures x and p read ∆ x = e − r and ∆ p = e r —reductionof noise in one quadrature is accompanied by an addednoise in the other one. This may be conveniently visual-ized in the phase-space picture by error disks represent-ing uncertainty in quadratures in different directions, seeFig. 1. In such a representation squeezed states corre-spond to ellipses while coherent states are representedby circles. Importantly, the fact that the uncertainty ofone of the quadratures can be less relatively to the othermakes it possible to design an interferometric schemewhere the measured photon number fluctuations are be-low that of a coherent state and allows for a sub-shotnoise phase estimation precision, see Sec. III. As squeezedstates in general cannot be described as mixtures of co-herent states, they are non-classical and their featurescannot be fully described by classical electrodynamics.Nevertheless, they can be relatively easily prepared usingnon-linear optical elements in the process of parametricdown conversion (Bachor and Ralph, 2004).The special type of squeezed states which is most rel-evant from the metrological perspective is the class ofthe squeezed vacuum states that possess vanishing meanvalues of their quadratures, i.e. h ˆ z i = 0 : | r i = ˆ S ( r ) | i . (11)In the Fock basis a squeezed vacuum state reads | r i = 1 √ cosh r ∞ X n =0 H n (0) √ n ! (cid:16) tanh r (cid:17) n e i n θ | n i , (12)where H n (0) denotes values of n -th Hermite polynomialat x = 0 . As for odd n H n ( x ) is antisymmetric andthus H n (0) = 0 , it follows that squeezed vacuum statesare superpositions of Fock states with only even photonnumbers. The average number of photons in a squeezedvacuum state is given by h n i = sinh r , what means that,despite their name, a squeezed vacuum states containphotons, possibly a lot of them.
3. Two-mode squeezed states
The simplest non-classical two-mode Gaussian stateis the so-called two-mode squeezed vacuum state or the twin-beam state (Walls and Milburn, 1995). Mathemat-ically, such a state is generated from the vacuum by atwo-mode squeezing operation, so that | ξ i = ˆ S ( ξ ) | , i , (13)where ˆ S ( ξ ) = exp( ξ ∗ ˆ a ˆ b − ξ ˆ a † ˆ b † ) and ξ = | ξ | e i θ , whereasin the Fock basis it reads | ξ i = 1cosh ξ ∞ X n =0 ( − n e i θ tanh n ξ | n, n i . (14)A notable feature of the twin-beam state, which may beclearly seen from Eq. (14), is that it is not a product ofsqueezed states in modes a and b , but rather it is cor-related in between them being a superposition of termswith the same number of photons in both modes. Its firstmoments of all quadratures are zero, h ˆ z i = 0 , whereas inthe case of ξ = | ξ | its covariance matrix has a particularlysimple form: σ = cosh(2 ξ ) 0 sinh(2 ξ ) 00 cosh(2 ξ ) 0 − sinh(2 ξ )sinh(2 ξ ) 0 cosh(2 ξ ) 00 − sinh(2 ξ ) 0 cosh(2 ξ ) . (15)Such a covariance matrix clearly indicates the presenceof correlations between the modes and since the state(13) is pure this implies immediately the presence of the mode-entanglement . In fact, in the limit of large squeez-ing coefficient | ξ | → ∞ such twin-beam state becomes theoriginal famous Einstein-Podolsky-Rosen state (Adesso,2006; Banaszek and Wódkiewicz, 1998) that violates as-sumptions of any realistic local hidden variable theory.Twin-beam states may be generated in a laboratory byvarious non-linear processes such as four- and three-wavemixing (Bachor and Ralph, 2004; Reid and Drummond,1988). Alternatively, they may be produced by mixingtwo single-mode squeezed vacuum states with oppositesqueezing angles on a fifty-fifty beam-splitter.
4. General two-mode Gaussian states
General two-mode Gaussian state is rather difficult towrite in the Fock basis, so it is best characterized by its × real symmetric covariance matrix σ = (cid:18) [ σ ] [ σ ][ σ ] [ σ ] (cid:19) , (16)where σ ij represent blocks with × matrices de-scribing correlations between the i -th and the j -thmode, and the vector of the first moments h ˆ z i = {h ˆ x i , h ˆ p i , h ˆ x i , h ˆ p i} . This in total gives up to four-teen real parameters describing the state: ten covari-ances, two displacement amplitudes and two phases ofdisplacement. General Gaussian states are in principlefeasible within current technological state of art, as anypure Gaussian state can theoretically be generated fromthe vacuum by utilizing only a combination of one-, two-mode squeezing and displacement operations with helpof beam-splitters and one-mode rotations (Adesso, 2006).Furthermore, mixed Gaussian states are obtained as a re-sult of tracing out some of the system degrees of freedom,which is effectively the case in the presence of light losses,or by adding a Gaussian noise to the state. C. Definite photon number states
Gaussian states are important from the practical pointof view due to the relative ease with which their may beprepared. From a conceptual point of view, however,when asking fundamental questions on limits to quan-tum enhancement in interferometry, states with a defi-nite photon number prove to be a better choice. Themain reason is that photons are typically regarded as aresource in interferometry and when benchmarking dif-ferent interferometric schemes it is natural to restrict theclass of states with the same number of photons, i.e. thesame resources consumed. A general M -mode state con-sisting of N photons is given by: ρ N = X | n | = | n ′ | = N ρ n , n ′ | n ih n ′ | , (17)where | n | = P i n i , so that the summation is restrictedonly to terms with exactly N photons in all the modes.Apart from the vacuum state | i no Gaussian state fallsinto this category.States with an exact photon number are extensivelyused in other fields of quantum information processing,including quantum communication and quantum com-puting (Kok et al. , 2007; Pan et al. , 2012). Most of thequantum computation and communication schemes aredesigned with such states in mind, as they provide themost intuitive and clear picture of the role the quantumfeatures play in these tasks. For large N , however, stateswith a definite photon numbers are notoriously hard toprepare and states with N of the order of can only beproduced with the present technology pushed to its limits(Hofheinz et al. , 2008; Sayrin et al. , 2011; Torres et al. ,2011). When considering states with definite N , it is alsopossible to easily switch between the mode- and particle-description of the states of light, which is a feature thatwe discuss in the following section. D. Particle description
When dealing with states of definite photon number,instead of thinking about modes as quantum subsystemsthat possess some number of excitations (photons), wemay equivalently consider the “first quantization” formal-ism and regard photons themselves as elementary sub-systems. Fundamentally, photons are indistinguishableparticles and since they are bosons their wave functionshould always be permutation-symmetric. Still, it is com-mon in the literature to use a description in which pho-tons are regarded as distinguishable particles and adopta notation such that | m i = | m i ⊗ | m i ⊗ · · · ⊗ | m N i N (18)denotes a product state of N photons, where the i -th pho-ton occupies the mode m i . We explicitly add subscriptsto the kets above labeling each constituent photon, inorder to distinguish this notation from the mode descrip-tion of Eq. (1), where kets denoted various modes and notthe distinct particles. The description (18) is legitimateprovided there are some degrees of freedom that ascribe ameaning to the statement “the i -th photon” . For example,in the case when photons are prepared in non-overlappingtime-bins, the time-bin degree of freedom plays the roleof the label indicating a particular photon, whereas thespatial characteristics determine the state of a given pho-ton. Nevertheless, if we assume the overall wave functiondescribing also the temporal degrees of freedom of thecomplete state to be fully symmetric, the notion of the“ i -th photon” becomes meaningless.A general pure state of N “distinguishable” photonshas the form: | ψ N i = X m c m | m i (19)where P m | c m | = 1 . If indeed there is no additional de-gree of freedom that makes the notion of “the i -th photon”meaningful, the above state should posses the symmetryproperty such that c m = c Π( m ) , where Π is an arbitrarypermutation of the N indices.Consider for example a Fock state | n i = | n i . . . | n M i of N = n + · · · + n M indistinguishable photons in M modes. In the particle description the state has the form: | n i = r n ! . . . n M ! N ! × X Π | Π( { , . . . , | {z } n , . . . , | {z } n , . . . , M, . . . , M | {z } n M } ) i , (20)where the sum is performed over all non-trivial permuta-tions Π of the indices inside the curly brackets (Shankar,1994). Since all quantum states may be written in theFock basis representation, by the above construction onecan always translate any quantum state to the particledescription. E. Mode vs particle entanglement
One of the most important features which makes thequantum theory different from the classical one is thenotion of entanglement (Horodecki et al. , 2009). Thisphenomenon plays also an important role in quantummetrology and is often claimed to be the crucial resourcefor the enhancement of the measurement precision (Gio-vannetti et al. , 2006; Pezzé and Smerzi, 2009). Conflict-ing statements can be found in the literature, however,as some of the authors claim that entanglement is notindispensable to get a quantum precision enhancement(Benatti and Braun, 2013). This confusion stems simplyfrom the fact that entanglement is a relative concept de-pendent on the way we divide the relevant Hilbert spaceinto particular subsystems. In order to clarify these is-sues, it is necessary to explicitly study relation betweenmode- and particle-entanglement, i.e. entanglement withrespect to different tensor product structures used in thetwo descriptions.Firstly, let us go through basic definitions and notionsof entanglement. The state ρ AB of two parties A and B is called separable if and only if one can write it as amixture of product of states of individual subsystems: ρ ( AB ) = X i p i ρ ( A ) i ⊗ ρ ( B ) i , p i ≥ . (21)Entangled states are defined as all states that are notseparable. A crucial feature of entanglement is that itdepends on the division into subsystems. For example,consider three qubits A , B and C and their joint quantumstate ρ ( ABC ) = P i,j =0 12 | i ih j |⊗| i ih j |⊗| ih | . This state isseparable with respect to the AB | C cut but is entangledwith respect to the A | BC cut.As a first example, consider a two-mode Fock state | i a | i b which represents one photon in mode a and onephoton in mode b . This state written in the mode formal-ism of Eq. (1) is clearly separable. On the other hand,photons are indistinguishable bosons and if we would liketo write their state in the particle formalism of Eq. (18)we have to symmetrize over all possible permutations ofparticles, thus obtaining the state | i a | i b = 1 √ | a i | b i + | b i | a i ) . (22)In this representation the state is clearly entangled. Wemay thus say that the state contains particle entangle-ment but not the mode entanglement.If, however, we perform the Hong-Ou-Mandel exper-iment and send the | i a | i b state through a balancedbeam-splitter which transforms mode annihilation oper-ators as ˆ a → (ˆ a + ˆ b ) / √ , ˆ b → (ˆ a − ˆ b ) / √ , the resultingstate reads: | i a | i b → √ | i a | i b −| i a | i b ) = 1 √ | a i | a i −| b i | b i ) , (23) which is both mode- and particle-entangled. Mode en-tanglement emerges because the beam-splitter is a jointoperation over two modes that introduces correlationsbetween them. On the other hand, it is a local opera-tion with respect to the particles, i.e. it can be written as U ⊗ U in the particle representation, and does not cou-ple photons with each other. Thus, using a beam-splitterone may change mode entanglement but not the contentof particle entanglement.As a second example, consider two modes of light, a and b , each of them in coherent state with the same am-plitude α , | α i a | α i b . This state clearly has no mode en-tanglement. Since this state does not have a definitephoton number, in order to ask questions about the par-ticle entanglement we first need to consider its projectionon one of the N -photon subspaces—one can think of anon-demolition total photon-number measurement yield-ing result N . After normalizing the projected state weobtain: | ψ N i = [ | α i a | α i b ] ( N ) = 1 √ N N X n =0 s(cid:18) Nn (cid:19) | n i a | N − n i b , (24)which in the particle representation reads: | ψ N i = 1 √ N N O i =1 ( | a i i + | b i i ) (25)and is clearly a separable state. The fact that productsof coherent states contain no particle entanglement is inagreement with our definition of classical state given inEq. (9) being a mixture of products of coherent states.A classical state according to this definition will contain neither mode nor particle entanglement.As a last example, consider the case of particular inter-est for quantum interferometry, i.e. a coherent state ofmode a and a squeezed vacuum state of mode b : | α i a | r i b .Again this state has no mode entanglement. On the otherhand it is particle entangled. To see this, consider e.g. thetwo-photon sector, which up to irrelevant normalizationfactor reads: [ | α i| r i ] ( N =2) ∝ α | i a | i b + tanh r | i a | i b == α | a i | a i + tanh r | b i | b i (26)and contains particle entanglement provided both α and r are non-zero. We argue and give detailed argumentsin Sec. V that it is indeed the particle entanglement and not the mode entanglement that is relevant in quantum-enhanced interferometry. See also Killoran et al. (2014)for more insight into the relation between mode and par-ticle entanglement. III. MACH-ZEHNDER INTERFEROMETRY
We begin the discussion of quantum-enhancement ef-fects in optical interferometry by discussing the paradig- ϕ n a n b | α i| r i ϕ ab ab a ¢ b ¢ FIG. 2 The Mach-Zehnder interferometer, with two inputlight modes a , b and two output modes a ′ , b ′ . In a standardconfiguration a coherent state of light | α i is sent into mode a .In order to obtain quantum enhacement, one needs to makeuse of the b input port also, sending e.g. the squeezed vacuumstate | r i . matic model of the Mach-Zehnder (MZ) interferometer.We analyze the most popular interferometric schemes in-volving the use of coherent and squeezed states of lightaccompanied by a basic measurement-estimation proce-dure, in which the phase is estimated based on the valueof the photon-number difference between the two out-put ports of the interferometer. Such a protocol providesus with a benchmark that we may use in the followingsections when discussing the optimality of the interfer-ometry schemes both with respect to the states of lightused as well as the measurements and the estimation pro-cedures employed. A. Phase-sensing uncertainty
In the standard MZ configuration, depicted in Fig. 2,a coherent state of light is split on a balanced beam-splitter, the two beams acquire phases ϕ a , ϕ b respec-tively, interfere on the second beam-splitter and finallythe photon numbers n a , n b are measured at the outputports. Let ˆ a, ˆ b and ˆ a ′ , ˆ b ′ be the annihilation operators cor-responding to the two input and the two output modesrespectively. The combined action of the beam-splittersand the phase delays results in the effective transforma-tion of the annihilation operators: (cid:18) ˆ a ′ ˆ b ′ (cid:19) = 12 (cid:18) (cid:19) (cid:18) e iϕ a
00 e i ϕ b (cid:19) (cid:18) − i − i 1 (cid:19) (cid:18) ˆ a ˆ b (cid:19) == e i( ϕ a + ϕ b ) / (cid:18) cos( ϕ/ − sin( ϕ/ ϕ/
2) cos( ϕ/ (cid:19) (cid:18) ˆ a ˆ b (cid:19) , (27)where ϕ = ϕ b − ϕ a is the relative phase delay and forconvenience we assume that the beams acquire a − π/ or π/ phase when transmitted through the first or thesecond beam-splitter respectively. The common phasefactor e i( ϕ a + ϕ b ) / is irrelevant for further discussion inthis section and will be omitted.In order to get a better insight into the quantum-enhancement effects in the operation of the MZ interfer-ometer, it is useful to make use of the so-called Jordan-Schwinger map (Schwinger, 1965) and analyse the action of the MZ interferometer in terms of the algebra of theangular momentum operators (Yurke et al. , 1986). Letus define the operators: ˆ J x = 12 (ˆ a † ˆ b +ˆ b † ˆ a ) , ˆ J y = i2 (ˆ b † ˆ a − ˆ a † ˆ b ) , ˆ J z = 12 (ˆ a † ˆ a − ˆ b † ˆ b ) , (28)which fulfill the angular momentum commutation rela-tions [ ˆ J i , ˆ J j ] = i ǫ ijk ˆ J k while the corresponding square ofthe total angular momentum reads: ˆ J = ˆ N ˆ N ! , ˆ N = ˆ a † ˆ a + ˆ b † ˆ b, (29)where ˆ N is the total photon number operator. The actionof linear optical elements appearing in the MZ interfer-ometer can now be described as rotations in the abstractspin space: ˆ a ′ = U ˆ aU † , ˆ b ′ = U ˆ bU † , U = exp( − i α ˆ J · s ) ,where ˆ J = { ˆ J x , ˆ J y , ˆ J z } and α , s are the angle and theaxis of the rotation respectively. In particular, the bal-anced beam splitter is a rotation around the x axis by anangle π/ : U = exp( − i π ˆ J x ) , while the phase delay is a ϕ rotation around the z axis: U = exp( − i ϕ ˆ J z ) . Insteadof analysing the transformation of the annihilation oper-ators, it is more convenient to look at the correspondingtransformation of the J i operators themselves: ˆ J ′ x ˆ J ′ y ˆ J ′ z = − cos ϕ − sin ϕ ϕ cos ϕ
00 0 1 ×× −
10 1 0 ˆ J x ˆ J y ˆ J z = cos ϕ ϕ − sin ϕ ϕ ˆ J x ˆ J y ˆ J z , (30)which makes it clear that the sequence of π/ , ϕ and − π/ rotations around axes x , z and x respectively, re-sults in an effective ϕ rotation around the y axis.Using the above formalism, let us now derive a sim-ple formula for uncertainty of phase-sensing based on themeasurement of the photon-number difference at the out-put. Note that ˆ n a − ˆ n b = 2 ˆ J z , so the photon-numberdifference measurement is equivalent to the ˆ J z measure-ment. Utilizing Eq. (30) in the Heisenberg picture, theaverage J z evaluated on the interferometer output statemay be related to the average of J ′ z of the input state | ψ i in as h ˆ J z i = cos ϕ h ˆ J z i in − sin ϕ h ˆ J x i in . (31)In order to assess the precision of ϕ -estimation, we alsocalculate the variance of the ˆ J z operator of the outputstate of the interferometer: ∆ J z = cos ϕ ∆ J z | in + sin ϕ ∆ J x | in + − ϕ cos ϕ cov ( J x , J z ) | in , (32)0where cov ( J x , J z ) = h ˆ J x ˆ J z + ˆ J z ˆ J x i − h ˆ J x ih ˆ J z i is thecovariance of the two observables. The precision of es-timating ϕ can now be quantified via a simple error-propagation formula: ∆ ϕ = ∆ J z (cid:12)(cid:12)(cid:12)(cid:12) d h ˆ J z i d ϕ (cid:12)(cid:12)(cid:12)(cid:12) . (33) B. Coherent-state interferometry
Let us now analyze the precision given by Eq. (33)for the standard optical interferometry with the inputstate | ψ i in = | α i| i , representing a coherent state and nolight at all being sent into the input modes a and b of theinterferometer in Fig. 2. The relevant quantities requiredfor calculating the precision given in Eq. (33) read: h ˆ J z i in = 12 | α | , h ˆ J x i in = 0 , ∆ J z | in = ∆ J x | in = 14 | α | , cov ( J x , J z ) | in = 0 (34)yielding the precision: ∆ ϕ | α i| i = | α | | α | | sin ϕ | = 1 | α sin ϕ | = 1 p h N i| sin ϕ | , (35)where the average photon number h N i = D ˆ N E = | α | .The above formula represents / p h N i shot noise scal-ing of precision characteristic for the classical inter-ferometry. The shot noise is a consequence of the ∆ J z effectively representing the Poissonian fluctuationsof the photon-number difference measurements at theoutput ports. Yet, although such fluctuations are ϕ -independent, the average photon-number difference h J z i changes with ϕ with speed proportional to | sin ϕ | ap-pearing in Eq. (35), so that the optimal operating pointsare at ϕ = π/ , π/ . C. Fock state interferometry
We can attempt to reduce the estimation uncertaintyusing more general states of light at the input. For ex-ample, we can replace the coherent state with an N -photon Fock state, so that | ψ i in = | N i| i . This is aneigenstate of ˆ J z and hence ∆ J z | in = 0 , and only the ∆ J x | in contributes to the ˆ J z variance at the output: ∆ J z = N sin ϕ . Since h ˆ J z i = N cos ϕ , the corre-sponding estimation uncertainty reads: ∆ ϕ | N i| i = 1 √ N , (36)being again shot-noise limited. The sole benefit of us-ing the Fock state is the lack of ϕ -dependence of the estimation precision. This, however, is scarcely of anyuse in practice, since one may always perform rough in-terferometric measurements and bring the setup close tothe optimal operating points before performing more pre-cise measurements there. Moreover, the ϕ -dependence inEq. (35) can be easily removed by taking into account notonly the photon-number difference observable ˆ J z but alsothe total photon number measured. By using their ratioas an effective observable in the r.h.s. of Eq. (33), or inother words by considering the “visibility observable”, theformula (35) is replaced by 36. D. Coherent + squeezed-vacuum interferometry
We thus need to use more general input states in orderto surpass the shot noise limit. Firstly, let us note thatsending the light solely to one of the input ports will notprovide us with the desired benefit. As in the end onlya photon-number measurement is assumed, which is notsensitive to any relative phase differences between variousFock terms of the output state, any scenario involving asingle-beam input may always be translated to the thesituation in which an incoherent mixture of Fock statesis sent onto the input port. Since the variance is a convexfunction with respect to state density matrices, and thusincreasing under mixing, such strategies are of no use forour purposes.Let us now consider a scheme were apart from thecoherent light we additionally send a squeezed-vacuumstate into the other input port (Caves, 1981): | ψ i in = | α i ⊗ | r i , see Fig. 2. This kind of strategy is be-ing implemented in current most advanced interferom-eters designed to detect gravitational waves like LIGO orGEO600 (LIGO Collaboration, 2011, 2013; Pitkin et al. ,2011). Assuming for simplicity that r is real, the relevantquantities required to calculate the estimation precisionread: h ˆ N i = | α | + sinh r, h J z i in = 12 ( | α | − sinh r ) , h J x i in = 0∆ J z | in = 14 (cid:0) | α | + sinh r (cid:1) , cov ( J x , J z ) | in = 0 , ∆ J x | in = 14 (cid:0) | α | cosh 2 r − Re ( α ) sinh 2 r + sinh r (cid:1) . (37)Hence, the usage of squeezed-vacuum as a second in-put allows to reduce the variance ∆ J x | in thanks tothe Re ( α ) sinh 2 r term above, which is then maximizedby choosing the phase of the coherent state such that α = Re ( α ) . This corresponds to the situation, when thecoherent state is displaced in phase space in the direc-tion in which the squeezed vacuum possesses its lowestvariance. With such an optimal choice of phase, substi-tuting the above formulas into Eq. (33), we obtain the1final expression for the phase-estimation precision: ∆ ϕ | α i| r i = q cot ϕ ( | α | + sinh r ) + | α | e − r + sinh r (cid:12)(cid:12) | α | − sinh r (cid:12)(cid:12) . (38)The optimal operation point are again clearly ϕ = π/ , π/ , since at them the first term under the squareroot, which is non-negative, vanishes. For a fair compar-ison with other strategies we should fix the total averagenumber of photons h N i , which is regarded as a resource,and optimize the split of energy between the coherentand the squeezed vacuum beams in order to minimize ∆ ϕ . This optimization can only be done numerically,but the solution can be well approximated analyticallyin the regime of h N i ≫ . In this regime the squeezedvacuum should carry approximately p h N i / of photons,so the squeezing factor obeys sinh r ≈ e r ≈ p h N i / while the majority of photons belongs to the coherentbeam. The resulting precision reads ∆ ϕ | α i| r i h N i≫ ≈ q h N i / (2 p h N i ) + p h N i / h N i − p h N i / h N i≫ ≈ h N i / (39)and proves that indeed this strategy offers better thanshot noise scaling of precision.While the above example shows that indeed quantumstates of light may lead to an improved sensitivity, theissue of optimality of the proposed scheme has not beenaddressed. In fact, keeping the measurement-estimationscheme unchanged, it is possible to further reduce theestimation uncertainty by sending a more general two-mode Gaussian states of light with squeezing present inboth input ports and reach the ∝ / h N i scaling of pre-cision (Olivares and Paris, 2007; Yurke et al. , 1986). Weskip the details here, as the optimization of the generalGaussian two-mode input state minimizing the estima-tion uncertainty of the scheme considered is cumbersome.More importantly, using the tools of estimation theoryintroduced in Sec. IV, we will later show in Sec. V thateven with | α i ⊗ | r i class of input states it is possible toreach the ∝ / h N i scaling of precision, but this requiresa significantly modified measurement-estimation scheme(Pezzé and Smerzi, 2008) and different energy partitionbetween the two input modes. E. Definite photon-number state interferometry
Even though definite photon-number states are techni-cally difficult to prepare, they are conceptually appealingand we make use of them to demonstrate explicitly thepossibility of achieving the /N Heisenberg scaling of es-timation precision in Mach-Zehnder interferometry. Wehave already shown that sending an N -photon state intoa single input port of the interferometer does not lead to an improved precision compared with a coherent-state–based strategy. Therefore, we need to consider stateswith photons being simultaneously sent into both inputports. A general N -photon two-mode input state can bewritten down using the angular momentum notation as | ψ i in = j X m = − j c m | j, m i , (40)where j = N/ and | j, m i = | j + m i| j − m i in the standardmode-occupation notation. In particular, | j, j i = | N i| i corresponds to a state with all the photons being sentinto the upper input port. One can easily check thatangular momentum operators introduced in Eq. (28) actin a standard way on the | j, m i states. For concretenessassume N is even and consider the state (Yurke et al. ,1986): | ψ i in = 1 √ | j, i + | j, i ) == 1 √ (cid:0)(cid:12)(cid:12) N (cid:11) (cid:12)(cid:12) N (cid:11) + (cid:12)(cid:12) N + 1 (cid:11) (cid:12)(cid:12) N − (cid:11)(cid:1) , (41)for which: h ˆ J z i in = 12 , h ˆ J x i in = 12 p j ( j + 1) , ∆ J z | in = 12 , ∆ J x | in = 12 j ( j + 1) − , cov ( J x , J z ) | in = 0 . (42)Plugging the above expressions into Eq. (33), we get: ∆ ϕ = q cos ϕ + sin ϕ [ j ( j + 1) − | sin ϕ + cos ϕ p j ( j + 1) | . (43)The optimal operation point corresponds to sin ϕ = 0 ,where we benefit from large h ˆ J x i in and low ∆ J z | in mak-ing the state very sensitive to rotations around the y axis.The resulting precision reads: ∆ ϕ = 1 p j ( j + 1) N ≫ ≈ N , (44)indicating the possibility of achieving the Heisenbergscaling of precision. We should note here, that a sim-pler state | ψ i in = | j, i = | N/ i| N/ i called the twin-Fock state where N/ photons are simultaneously sentinto each of the input ports, is also capable of providingthe Heisenberg scaling of precision (Holland and Burnett,1993), but requires a different measurement-estimationscheme which goes beyond the analysis of the averagephoton-number difference at the output, see Sec. V.C.3.One should bear in mind that the expressions for at-tainable precision presented in this section base on a sim-ple error propagation formula calculated at a particularoperating point. Therefore, in order to approach any of2 ϕ ϕ θ FIG. 3 Other popular two-input/two-output mode interfer-ometers: (a) Michelson interferometer with one-way phasedelays ϕ a , ϕ b in the respective arms, (b) Fabry-Pérrot inter-ferometer with one-way phase delay θ and power transmission T of the mirrors. the precisions claimed, one needs to first lock the interfer-ometer to operate close to an optimal point, which alsorequires some of the resources to be consumed. Rigor-ous quantification of the total resources needed to attaina given estimation precision starting with a completelyunknown phase may be difficult in general. We return tothis issue in Sec. V, where we are able to resolve this prob-lem by approaching it with the language of the Bayesianinference. F. Other interferometers
Even though we have focused on the Mach-Zehnder in-terferometer setup, analogous results could be obtainedfor other optical interferometric configurations, such asthe Michelson interferometer, the Fabry-Pérrot inter-ferometer, as well as the atomic interferometry setupsutilized in: atomic clocks operation (Diddams et al. ,2004), spectroscopy (Leibfried et al. , 2004), magnetome-try (Budker and Romalis, 2007) or the BEC interferom-etry (Cronin et al. , 2009). We briefly show below thatdespite physical differences the mathematical frameworkis common to all these cases and as such the results pre-sented in this review, even though derived with the sim-ple optical interferometry in mind, have a much broaderscope of applicability.The Michelson interferometer is depicted in Fig. 3a.Provided the output modes a ′ , b ′ can be separated fromthe input ones a , b via an optical circulator, the Michel-son interferometer is formally equivalent to the Mach-Zehnder interferometer. The input output relations areidentical as in Eq. (27) with ϕ = 2( ϕ b − ϕ a ) , as the lightacquires the relative phase twice—traveling both to andfrom the end mirrors.Consider now the Fabry-Pérrot interferometer de-picted in Fig. 3b. We assume for simplicity that bothmirrors have the same power transmission coefficient T and the phase θ is acquired while the light travels fromone mirror to the other inside the interferometer. The resulting input-output relation reads: (cid:18) ˆ a ′ ˆ b ′ (cid:19) = 1(2 − T )i sin θ − T cos θ ×× (cid:18) √ − T sin θ TT √ − T sin θ (cid:19)(cid:18) ˆ a ˆ b (cid:19) (45)and up to an irrelevant global phase may be rewritten as: (cid:18) ˆ a ′ ˆ b ′ (cid:19) = (cid:18) cos ϕ/ ϕ/
2i sin ϕ/ ϕ/ (cid:19) (cid:18) ˆ a ˆ b (cid:19) ,ϕ = 2 arcsin (cid:18) T √ T +4(1 − T ) sin θ (cid:19) . (46)In terms of the angular momentum operators the abovetransformation is simply a ϕ -rotation around the x axis.Thus, up to a change of the rotation axis, the actionof the Fabry-Pérrot interferometer with phase delay θ is equivalent to the one of the MZ interferometer withphase delay ϕ . Knowing the formulas for the estimationprecision of ϕ in the MZ interferometer, we can easilycalculate the corresponding estimation precision of θ viathe error propagation formula obtaining: ∆ θ = ∆ ϕ | ∂ϕ∂θ | = ∆ ϕ T + 4(1 − T ) sin θ T √ − T cos θ . (47)As a consequence, the above expression allows to trans-late all the results derived for the MZ interferometer tothe Fabry-Pérrot case.Ramsey interferometry is a popular technique for per-forming precise spectroscopic measurements of atoms. Itis widely used in atomic clock setups, where it allows tolock the frequency of an external source of radiation, ω ,to a selected atomic transition frequency, ω , between thesingle-atom excited and ground states, | e i and | g i (Did-dams et al. , 2004). In a typical Ramsey interferometricexperiment N atoms are initially prepared in the groundstate and are subsequently subjected to a π/ Rabi pulse,which transforms each of them into an equally weightedsuperposition of ground and excited states. Afterwards,they evolve freely for time t , before finally being sub-jected to a second π/ pulse, which in the ideal case of ω = ω would put all the atoms in the exited state. Incase of any frequency mismatch, the probability for anatom to be measured in an excited state is cos ( ϕ/ ,where ϕ = ( ω − ω ) t . Hence, by measuring the numberof atoms in the excited state, one can estimate ϕ andconsequently knowing t the frequency difference ω − ω .Treating two-level atoms as spin- / particles with theirtwo levels corresponding to up and down projections ofthe spin z component, we may introduce the total spinoperators ˆ J i = P Nk =0 ˆ σ ( k ) i , i = x, y, z , where ˆ σ ( k ) i arestandard Pauli sigma matrices acting on the k -th parti-cle. Evolution of a general input state can be writtenas: | ψ ϕ i = e − i ˆ J x π/ e i ˆ J z ϕ e − i ˆ J x π/ | ψ i in . (48)3 FIG. 4 Formal equivalence of Ramsey and MZ interferometry.The two atomic levels which are used in the Ramsey interfer-ometry play analogous roles as the two arms of the interfer-ometer, while the π/ pulses equivalent from a mathematicalpoint of view to the action of the beam-splitters. Quantum-enhancement in Ramsey interferometry may be obtained bypreparing the atoms in a spin-squeezed input state reducingthe variances of the relevant total angular momentum oper-ators, in a similar fashion as using squeezed states of lightleads to an improved sensitivity in the MZ interferometry. which is completely analogous to the MZ transforma-tion (30) with π/ pulses playing the role of the beam-splitters, see Fig. 4. The total spin z operator can bewritten as ˆ J z = 2(ˆ n g − ˆ n e ) , where ˆ n g , ˆ n e denote theground and excited state atom number operators respec-tively. Therefore, measurement of ˆ J z is equivalent tothe measurement of the difference of excited and non-excited atoms analogously to the optical case where itcorresponded to the measurement of photon-number dif-ference at the two output ports of the interferometer.Fluctuation of the number of atoms measured limits theestimation precision and in case of uncorrelated atomsis referred to as the projection noise , which may be re-garded as an analog of the optical shot noise. An im-portant difference from the optical case, though, is thatwhen dealing with atoms we are restricted to considerstates of definite particle-number. Thus, there is no exactanalogue of coherent or squeezed states that we considerin the photonic case. We can therefore regard atomicRamsey interferometry as a special case of the MZ in-terferometry with inputs restricted to states of definiteparticle-number, discussed in Sec. III.E, and further re-late the precision of estimating the frequency differenceto the precision of phase estimation via ∆ ω = ∆ ϕt . (49)Beating the projection noise requires the input state ofthe atoms to share some particle entanglement. From anexperimental point of view the most promising class ofstates are the so-called one-axis or two-axis spin-squeezedstates , which may be realized in BEC and atomic sys-tems interacting with light (Kitagawa and Ueda, 1993;Ma et al. , 2011). In fact, these states may be regarded asa definite particle-number analogues of optical squeezed states. In particular, starting with atoms in a groundstate the two-axis spin-squeezed states may be obtainedvia: | ψ χ i = e − χ ( ˆ J − ˆ J − ) | g i ⊗ N , (50)where ˆ J ± = ˆ J x ± i ˆ J y . The above formula resembles thedefinition of an optical squeezed state given in Eq. (10),where ˆ a , ˆ a † operators are replaced by ˆ J − , ˆ J + . With anappropriate choice of squeezing strength χ as a functionof N it is possible to achieve the Heisenberg scaling ofprecision ∆ ω ∝ /N (Ma et al. , 2011; Wineland et al. ,1994).Analogous schemes may also be implemented in BEC(Cronin et al. , 2009; Gross et al. , 2010). In particu-lar, BEC opens a way of realizing a specially appealingmatter-wave interferometry, in which, similarly to the op-tical interferometers, the matter-wave is split into twospatial modes, evolves and finally interferes resulting ina spatial fringe pattern that may be used to estimate therelative phase acquired by the atoms (Shin et al. , 2004).Such a scheme may potentially find applications in pre-cise measurements of the gravitational field (Andersonand Kasevich, 1998). Yet, in this case, the detectioninvolves measurements of positions of the atoms form-ing the interference fringes, what makes the estimationprocedure more involved than in the simple MZ scheme(Chwedeńczuk et al. , 2012, 2011), but eventually the pre-cisions for the optimal estimation schemes should coin-cide with the ones obtained for the MZ interferometry.Finally, we should also mention that atomic ensemblesinteracting with light are excellent candidates for ultra-precise magnetometers (Budker and Romalis, 2007). Col-lective magnetic moment of atoms rotates in the pres-ence of magnetic field to be measured, what again can beseen as an analogue of the MZ transformation on ˆ J i inEq. (30). The angle of the atomic magnetic-moment rota-tion is determined by sending polarized light which due tothe Faraday effect is rotated proportionally to the atomicmagnetic-moment component in the direction of the lightpropagation. In standard scenarios, the ultimate pre-cision will be affected by both the atomic projectionnoise, due to characteristic uncertainties of the collec-tive magnetic-moment operator for uncorrelated atoms,and the light shot noise. The quantum enhancement ofprecision may again be achieved by squeezing the atomicstates (Sewell et al. , 2012; Wasilewski et al. , 2010) aswell as by using the non-classical states of light (Horrom et al. , 2012), what in both cases allows to go beyond theprojection and shot-noise limits. IV. ESTIMATION THEORY
In this section we review the basics of both classicaland quantum estimation theory. We present Fisher In-formation and Bayesian approaches to determining the4optimal estimation strategies and discuss tools particu-larly useful for analysis of optical interferometric setups.
A. Classical parameter estimation
The essential question that has been addressed bystatisticians long before the invention of quantum me-chanics is how to most efficiently extract informationfrom a given data set, which is determined by some non-deterministic process (Kay, 1993; Lehmann and Casella,1998).In a typical scenario we are given an N -point data set x = { x , x , . . . , x N } which is a realization of N indepen-dent identically distributed random variables, X N , eachdistributed according to a common Probability DensityFunction (PDF), p ϕ ( X ) , that depends on an unknownparameter ϕ we wish to determine. Our goal is to con-struct an estimator ˜ ϕ N ( x ) that should be interpreted asa function which outputs the most accurate estimate ofthe parameter ϕ based on a given data set. Importantly,as the estimator ˜ ϕ N is build on a sample of random data,it is a random variable itself and the smaller are its fluc-tuations around the true value ϕ the better it is.Typically, two approaches to the problem of the choiceof the optimal estimator are undertaken. In the so-calledfrequentist or classical approach, ϕ is assumed to be adeterministic variable with an unknown value that, ifknown, could in principle be stated to any precision. Inthis case, one of the basic tools in studying optimal esti-mation strategies is the Fisher information, and hencewe will refer to this approach as the Fisher informa-tion approach. In contrast, when following the
Bayesian paradigm, the estimated parameter is a random variableitself that introduces some intrinsic error that accountsfor the lack of knowledge about ϕ we possess prior toperforming the estimation. We describe both approachesin detail below.
1. Fisher Information approach
In this approach p ϕ ( X ) is regarded as a family of PDFsparametrized by ϕ —the parameter to be estimated basedon the registered data x . The performance of a givenestimator ˜ ϕ N ( x ) is quantified by the Mean Square Error(MSE) deviation from the true value ϕ : ∆ ˜ ϕ N = D ( ˜ ϕ N ( x ) − ϕ ) E = ˆ d N x p ϕ ( x ) ( ˜ ϕ N ( x ) − ϕ ) . (51)A desired property for an estimator is that it is unbiased : h ˜ ϕ N i = ˆ d N x p ϕ ( x ) ˜ ϕ N ( x ) = ϕ , (52)so that on average it yields the true parameter value.The optimal unbiased estimator is the one that minimizes ∆ ϕ N for all ϕ . Looking for the optimal estimator maybe difficult and it may even be the case that there is nosingle estimator that minimizes the MSE for all ϕ .Still, one may always construct the so-called Cramer-Rao Bound (CRB) that lower-bounds the MSE of anyunbiased estimator ˜ ϕ N (see e.g. (Kay, 1993) for a re-view): ∆ ˜ ϕ N ≥ N F [ p ϕ ] , (53)where F is the Fisher Information (FI), and can be ex-pressed using one of the formulas below: F [ p ϕ ] = ˆ d x p ϕ ( x ) (cid:20) ∂ p ϕ ( x ) ∂ϕ (cid:21) == *(cid:18) ∂∂ϕ ln p ϕ (cid:19) + = − (cid:28) ∂ ∂ϕ ln p ϕ (cid:29) , (54)The basic intuition is that the bigger the FI is thehigher estimation precision may be expected. The FIis non-negative and additive for uncorrelated events, sothat F h p (1 , ϕ i = F h p (1) ϕ i + F h p (2) ϕ i , for p (1 , ϕ ( x , x ) = p (1) ϕ ( x ) p (2) ϕ ( x ) and in particular: F (cid:2) p Nϕ (cid:3) = N F [ p ϕ ] ,which can be easily verified using the last expression indefinition (54). The FI is straightforward to calculateand once an estimator is found that saturates the CRBit is guaranteed to be optimal. In general estimators sat-urating the CRB are called efficient . The sufficient andnecessary condition for efficiency is the following condi-tion on the PDF and the estimator (Kay, 1993): ∂∂ϕ ln p ϕ ( x ) = N F [ p ϕ ] ( ˜ ϕ k ( x ) − ϕ ) . (55)An estimator ˜ ϕ satisfying the above equality exist onlyfor a special class of PDFs belonging to the so called exponential family of PDFs, for which: ln p ϕ ( x ) = a ( ϕ ) + b ( x ) + c ( ϕ ) d ( x ) , a ′ ( ϕ ) c ′ ( ϕ ) = − ϕ, (56)where a ( ϕ ) , c ( ϕ ) and b ( x ) , d ( x ) are arbitrary functionsand primes denote differentiation over ϕ . In general,however, the saturability condition cannot be met.Note that in general FI is a function of ϕ , so thatdepending on the true value of the parameter, the CRBputs weaker or stronger constraints on the minimal MSE.Actually, one is not always interested in the optimal esti-mation strategy that is valid globally —for any potentialvalue of ϕ —but may want to design a protocol that worksoptimally for ϕ confined to some small parameter range.In this case one can take a local approach and analyzethe CRB at a given point ϕ = ϕ . Formally derivationof the CRB at a given point requires only a weaker localunbiasedness condition: ∂∂ϕ h ˜ ϕ N i (cid:12)(cid:12)(cid:12)(cid:12) ϕ = ϕ = 1 (57)5at a given parameter value ϕ . FI at ϕ is a local quantitythat depends only on p ϕ ( X ) | ϕ = ϕ and ∂ p ϕ ( X ) ∂ϕ (cid:12)(cid:12)(cid:12) ϕ = ϕ , as ex-plicitly stated in Eq. (54). As a result, the FI is sensitiveto changes of the PDF of the first order in δϕ = ϕ − ϕ .Looking for the optimal locally unbiased estimator at agiven point ϕ makes sense provided one has a substan-tial prior knowledge that the true value of ϕ is close to ϕ . This may be the case if the data is obtained froma well controlled physical system subjected to small ex-ternal fluctuations or if some part of the data had beenused for preliminary estimation narrowing the range ofcompatible ϕ to a small region around ϕ . In this case,even if the condition for saturability of the CRB cannotbe met, it still may be possible to find a locally unbiasedestimator which will saturate the CRB at least at a givenpoint ϕ . The explicit form of the estimator can easilybe derived from Eq. (55) by substituting ϕ = ϕ : ˜ ϕ ϕ N ( x ) = ϕ + 1 N F [ p ϕ ] ∂ ln p ϕ ( x ) ∂ϕ (cid:12)(cid:12)(cid:12)(cid:12) ϕ = ϕ . (58)Fortunately, difficulties in saturating the CRB are onlypresent in the finite- N regime. In the asymptotic limit ofinfinitely many repetitions of an experiment, or equiva-lently, for an infinitely large sample, N → ∞ , a particularestimator called the Maximum Likelihood (ML) estima-tor saturates the CRB (Kay, 1993; Lehmann and Casella,1998). The ML estimator formally defined as ˜ ϕ ML N ( x ) = argmax ϕ p ϕ ( x ) (59)is a function that for a given instance of outcomes, x ,outputs the value of parameter for which this data sam-ple is the most probable. For finite N the ML estimatoris in general biased, but becomes unbiased asymptoti-cally lim N →∞ h ϕ ML N ( x ) i = ϕ and saturates the CR bound lim N →∞ N ∆ ˜ ϕ ML ( x ) = F [ p ϕ ] .
2. Bayesian approach
In this approach, the parameter to be estimated, ϕ ,is assumed to be a random variable that is distributedaccording to a prior PDF, p ( ϕ ) , representing the knowl-edge about ϕ one possesses before performing the estima-tion, while p ( x | ϕ ) denotes the conditional probability ofobtaining result x for parameter value ϕ . Notice, a sub-tle change in notation from p ϕ ( x ) in the FI approach to p ( x | ϕ ) in the Bayesian approach reflecting the change inthe role of ϕ which is a parameter in the FI approach anda random variable in the Bayesian approach. If we stickto the MSE as a cost function, we say that the estimator ˜ ϕ N ( x ) is optimal if it minimizes the average MSE (cid:10) ∆ ˜ ϕ N (cid:11) = ¨ dϕ d x p ( x | ϕ ) p ( ϕ ) ( ˜ ϕ N ( x ) − ϕ ) , (60) which, in contrast to Eq. (51), is also averaged over allthe values of the parameter with the Bayesian prior p ( ϕ ) .Making use of the Bayes theorem we can rewrite theabove expression in the form (cid:10) ∆ ˜ ϕ N (cid:11) = ˆ d x p ( x ) (cid:20) ˆ dϕ p ( ϕ | x ) ( ˜ ϕ N ( x ) − ϕ ) (cid:21) . (61)From the above formula it is clear that the optimal esti-mator is the one that minimizes terms in square bracketfor each x . Hence, we can explicitly derive the form ofthe Minimum Mean Squared Error (MMSE) estimator ∂∂ ˜ ϕ N ˆ dϕ p ( ϕ | x ) ( ˜ ϕ N ( x ) − ϕ ) = 0 = ⇒ ˜ ϕ MMSE N ( x ) = ˆ dϕ p ( ϕ | x ) ϕ = h ϕ i p ( ϕ | x ) (62)which simply corresponds to the average value of the pa-rameter computed with respect to the posterior PDF, p ( ϕ | x ) . The posterior PDF represents the knowledge wepossess about the parameter after inferring the informa-tion about it from the sampled data x . The correspond-ing MMSE reads: (cid:10) ∆ ˜ ϕ N (cid:11) = ˆ d x p ( x ) (cid:20) ˆ dϕ p ( ϕ | x ) (cid:16) ϕ − h ϕ i p ( ϕ | x ) (cid:17) (cid:21) = ˆ d x p ( x ) ∆ ϕ (cid:12)(cid:12) p ( ϕ | x ) , (63)so that it may be interpreted as the variance of the pa-rameter ϕ computed again with respect to the posteriorPDF p ( ϕ | x ) that is averaged over all the possible out-comes.The optimal estimation strategy within the Bayesianapproach depends explicitly on the prior PDF assumed.If either the prior PDF will be very sensitive to varia-tions of ϕ or the physical model will predict the datato be weakly affected by any parameter changes, so that p ( x | ϕ ) p ( ϕ ) ≈ p ( ϕ ) , the minimal ∆ ˜ ϕ N will be predomi-nantly determined by the prior distribution p ( ϕ ) and thesampled data will have limited effect on the estimationprocess. Therefore, it is really important in the Bayesianapproach to choose an appropriate prior PDF that, onone hand, should adequately represent our knowledgeabout the parameter before the estimation, but, on theother, its choice should not dominate the information ob-tained from the data.In principle, nothing prevents us to consider more gen-eral cost functions, C ( ˜ ϕ, ϕ ) , that in some situations maybe more suitable than the squared error. The correspond-ing optimal estimator will be the one that minimizes theaverage cost function h C i = ¨ dϕ d x p ( ϕ ) p ( x | ϕ ) C ( ˜ ϕ N ( x ) , ϕ ) . (64)In the context of optical interferometry the estimatedquantity of interest ϕ is the phase which is a circular6parameter, i.e. may be identified with a point on a circleor more formally as an element of the circle group U (1) and in particular ϕ ≡ ϕ + 2 nπ . Following (Holevo, 1982)the cost function should respect the parameter topology and the squared error is clearly not the proper choice.We require the cost function to be symmetric, C ( ˜ ϕ, ϕ ) = C ( ϕ, ˜ ϕ ) , group invariant , i.e. ∀ φ ∈ U (1) : C ( ˜ ϕ + φ, ϕ + φ ) = C ( ˜ ϕ, ϕ ) , and periodic , C ( ϕ + 2 nπ, ˜ ϕ ) = C ( ϕ, ˜ ϕ ) . Thisrestricts the class of cost functions to: C ( ˜ ϕ, ϕ ) = C ( δϕ ) = ∞ X n =0 c n cos[ n δϕ ] with δϕ = ˜ ϕ − ϕ . (65)Furthermore, we require C ( δϕ ) must rise monotonicallyfrom C (0) = 0 at δϕ = 0 to some C ( π ) = C max at δϕ = π ,so that C ′ ( δϕ ) ≥ , so that the coefficients c n must fulfillfollowing constraints: ∞ X n =0 c n = 0 , ∞ X n =0 ( − n c n = C max , ∞ X n =1 n c n ≤ , ∞ X n =1 n ( − n c n ≥ , (66)which may be satisfied by imposing ∀ n> : c n ≤ andtaking c n to decay at least quadratically with n . Lastly,for the sake of compatibility we would like the cost func-tion to approach the standard variance for small δϕ so that C ( δϕ ) = δϕ + O (cid:0) δϕ (cid:1) , which is equivalent to P ∞ n =1 n c n = − .In all the Bayesian estimation problems considered inthis work, we will consider the simplest cost function thatsatisfies all above-mentioned conditions with c = − c = 2 , ∀ n> : c n = 0 which reads explicitly: C ( ˜ ϕ, ϕ ) = 4 sin (cid:18) ˜ ϕ − ϕ (cid:19) . (67)Following the same argumentation as described inEq. (62) when minimizing the averaged MSE, one mayprove that for the above chosen cost function the aver-age cost h C i is minimized if an estimator, ˜ ϕ CN ( x ) , canbe found that for any possible data sample x collectedsatisfies the condition ˆ dϕ p ( ϕ | x ) sin (cid:0) ˜ ϕ CN ( x ) − ϕ (cid:1) = 0 . (68)
3. Example: Transmission coefficient estimation
In order to illustrate the introduced concepts let usconsider a simple model of parameter estimation, wherea single photon impinges on the beamsplitter with powertransmission and reflectivity equal respectively p and q = 1 − p . The experiment is repeated N times and basedon the data obtained—number of photons transmittedand the number of photons reflected—the goal is to es-timate the transmission coefficient p . This problem is equivalent to a coin-tossing experiment, where we assumean unfair coin, which “heads” and “tails” occurring withprobabilities p and q = 1 − p respectively.Probability that n out of N photons get transmittedis governed by the binomial distribution p N p ( n ) = (cid:18) Nn (cid:19) p n (1 − p ) N − n . (69)The FI equals F [ p N p ] = N/ [ p (1 − p )] and hence the CRBimposes a lower bound on the achievable estimation vari-ance: ∆ ˜ p N ≥ p (1 − p ) N . (70)Luckily the binomial probability distribution belongs tothe exponential family of PDFs specified in Eq. (56), andby inspecting saturability condition Eq. (55) it may beeasily checked, that the simple estimator ˜ p N ( n ) = n/N saturates the CRB. It is also worth mentioning that theoptimal estimator also coincides with the ML estimator,hence in this case the ML estimator is optimal also forfinite N and not only in the asymptotic regime.In the context of optical interferometry, we will dealwith an analogous situation, where photons are sent intoone input port of an interferometer and p , q correspondto the probabilities of detecting a photon in one of thetwo output ports. For a Mach-Zehnder interferometer,see Sec. III, the probabilities depend on the relative phasedelay difference between the arms of an interferometer ϕ : p = sin ( ϕ/ , q = cos ( ϕ/ , which is the actual param-eter of interest. Probability distribution as a function of ϕ then reads: p Nϕ ( n ) = (cid:18) Nn (cid:19) sin( ϕ/ n cos( ϕ/ N − n ) (71)and the corresponding FI and the CRB take the form F (cid:2) p Nϕ (cid:3) = N, ∆ ˜ ϕ N ≥ N . (72)Interestingly, FI does not depend on the actual value ϕ , what suggests that the achievable estimation preci-sion may be independent of the actual parameter value.However, for such a parametrization, the CRB satura-bility condition Eq. (55) does not hold and there is nounbiased estimator saturating the bound. Nevertheless,using Eq. (58), we may still write a locally unbiased esti-mator saturating CRB at ϕ : ˜ ϕ N ( n ) = ϕ − tan( ϕ /
2) +2 n/ ( N sin ϕ ) , which is possible provided sin ϕ = 0 .Since the CRB given in (72) can only be saturatedlocally it is worth looking at the MLE which we knowwill perform optimally in the asymptotic regime N → ∞ .Solving Eq. (59) we obtain ˜ ϕ N ( n ) = argmax ϕ ln p Nϕ ( n ) = ± arctan r nN − n (73)7and in general we notice that there are two equivalentmaxima. This ambiguity is simply the result of invarianceof the PDF p ϕ ( n ) with respect to the change ϕ → − ϕ and might have been expected. Hence in practice wewould need some additional information, possibly com-ing from prior knowledge or other observations, in orderto distinguish between this two cases and be entitled toclaim that the MLE saturates the CRB asymptoticallyfor all ϕ .We now analyze the same estimation problem employ-ing the Bayesian approach. Let us first consider the casewhere p is the parameter to be estimated and the rel-evant conditional probability p N ( n | p ) = p N p ( n ) is givenby Eq. (69). Choosing a flat prior distribution p ( p ) = 1 and the mean square as a cost function, we find usingEq. (62) and Eq. (63) that the MMSE estimator and thecorresponding minimal averaged MSE equal: ˜ p N ( n ) = n + 1 N + 2 , (cid:10) ∆ ˜ p N (cid:11) = 16 ( N + 2) , (74)which may be compared with the previously discussedFI approach where the optimal estimator was ˜ p ( n ) = n/N and the resulting variance when averaged over all p would yield h ∆ ˜ p N i = ´ d p p ( p − N = 1 / (6 N ) . Hence,in the limit N → ∞ these results converge to the onesobtained previously in the FI approach. This is a typicalsituation that in the case of large amount of data thetwo approaches yield equivalent results (van der Vaart,2000).We now switch to ϕ parametrization and consider p N ( n | ϕ ) = p Nϕ ( n ) as in Eq. (71). Assuming flat prior dis-tribution p ( ϕ ) = 1 / (2 π ) and the previously introducednatural cost function in the case of circular parameter C ( ˜ ϕ, ϕ ) = 4 sin [( ˜ ϕ − ϕ ) / , due to ± ϕ estimation ambi-guity we realize that the condition for the optimal esti-mator given in Eq. (68) is satisfied for a trivial estimator ˜ ϕ C ( n ) = 0 which does not take into account the mea-surement results at all. This can be understood oncewe realize that the ambiguity in the sign of estimatedphase ϕ and the possibility of estimating the phase withthe wrong sign is worse than not taking into accountthe measured data at all. In order to obtain a more in-teresting result we need to consider a subset of possiblevalues of ϕ over which the reconstruction is not ambigu-ous. If we choose ϕ ∈ [0 , π ) , and the corresponding prior p ( ϕ ) = 1 /π , Eq. (68) yields the optimal form of the es-timator and the corresponding minimal cost calculatedaccording to Eq. (64) are as follows: ˜ ϕ CN ( n ) = arctan (cid:16) f ( N,n ) (cid:17) (75) h C i = 2 − π ( N +1) N X n =0 p f ( N, n ) ! , (76)where f ( N, n ) = ( N − n )( n − / N − n − / /n !( N − n )! . Despite its complicated form the above formu- las simplify in the limit N → ∞ . The optimal es-timator approaches the ML estimator ˜ ϕ CN ( n ) ≈ × arctan p n/ ( N − n ) while the average cost function ap-proaches h C i ≈ /N indicating saturation of the CRBand confirming again that in the regime of many experi-mental repetitions the two discussed approaches coincide. B. Quantum parameter estimation
In a quantum estimation scenario the parameter ϕ is encoded in a quantum state ρ ϕ which is subject toa quantum measurement M x yielding measurement re-sult x with probability p ϕ ( x ) = Tr { ρ ϕ M x } . The esti-mation strategy is complete once an estimator function ˜ ϕ ( x ) is given ascribing estimated parameters to partic-ular measurement results. The quantum measurementmay be a standard projective von-Neumann measure-ment, M x M x ′ = M x δ x , x ′ , or a generalized measurement where the measurement operators form a Positive Oper-ator Valued Measure (POVM) with the only constraintsbeing M x ≥ , ´ d x M x = p ϕ ( x ) and their quantum mechanical origin becomes irrelevant.Hence, the estimation problem becomes then fully classi-cal and all the techniques developed in Sec. IV.A apply.
1. Quantum Fisher Information approach
The problem of determining the optimal measurementscheme for a particular estimation scenario is non-trivial.Fortunately analogously as in the classical estimation it isrelatively easy to obtain useful lower bounds on the min-imal MSE. The
Quantum Cramér-Rao Bound (QCRB)(Braunstein and Caves, 1994; Helstrom, 1976; Holevo,1982) is a generalization of the classical CRBs (53), whichlower bounds the variance of estimation for all possiblelocally unbiased estimators and the most general POVMmeasurements : ∆ ϕ N ≥ N F Q [ ρ ϕ ] with F Q [ ρ ϕ ] = Tr n ρ ϕ L [ ρ ϕ ] o (77) For clarity of notation, in what follows we drop the tilde symbolwhen writing estimator variance. F Q is the Quantum Fisher Information (QFI)while the Hermitian operator L [ ρ ϕ ] is the so called Sym-metric Logarithmic Derivative (SLD), which can be un-ambiguously defined for any state ρ ϕ via the relation ˙ ρ ϕ = ( ρ ϕ L [ ρ ϕ ]+ L [ ρ ϕ ] ρ ϕ ) . Crucially, QFI is solely deter-mined by the dependence of ρ ϕ on the estimated param-eter, and hence allows to analyze parameter sensitivityof given probe states without considering any particularmeasurements nor estimators. Explicitly, the SLD whenwritten in the eigenbasis of ρ ϕ = P i λ i ( ϕ ) | e i ( ϕ ) ih e i ( ϕ ) | reads: L [ ρ ϕ ] = X i,j h e i ( ϕ ) | ˙ ρ ϕ | e j ( ϕ ) i λ i ( ϕ ) + λ j ( ϕ ) | e i ( ϕ ) ih e j ( ϕ ) | , (78)where the sum is taken over the terms with non-vanishingdenominator. Analogously to the FI (54), the QFI is anadditive quantity when calculated on product states andin particular F Q (cid:2) ρ ⊗ Nϕ (cid:3) = N F Q [ ρ ϕ ] . Thus the N term inthe denominator of Eq. (77) may be equivalently inter-preted as the number of independent repetitions of anexperiment with a state ρ ϕ to form the data sample x of size N , or a single shot experiment with a multi-partystate ρ ⊗ Nϕ .Crucially, as proven in (Braunstein and Caves, 1994;Nagaoka, 2005), there always exist a measurementstrategy—a projection measurement in the eigenbasis ofthe SLD—for which the FI calculated for the resultingprobability distribution equals the QFI, and consequentlythe bounds (53) and (77) coincide. Hence the issue of sat-urability of the QCRB amounts to the problem of satura-bility of the corresponding classical CRB. As discussedin detail in Sec. IV.A.1, the bound is therefore globallysaturable for a special class of probability distributionsbelonging to the so called exponential family, and if thisis not the case the saturability is achievable either inthe asymptotic limit of many independent experiments N → ∞ or in the local approach when one estimatessmall fluctuation of the parameter in the vicinity of aknown value ϕ .For pure states, ρ ϕ = | ψ ϕ ih ψ ϕ | , the QFI in Eq. (77)simplifies to F Q [ | ψ ϕ i ] = 4 (cid:18)D ˙ ψ ϕ | ˙ ψ ϕ E − (cid:12)(cid:12)(cid:12)D ˙ ψ ϕ | ψ ϕ E(cid:12)(cid:12)(cid:12) (cid:19) , | ˙ ψ ϕ i = d | ψ ϕ i d ϕ . (79)Yet, for general mixed states, calculation of the QFI in-volves diagonalization of the quantum state ρ ϕ , in or-der to calculate the SLD, and becomes tedious for probestates living in highly dimensional Hilbert spaces.Interestingly, the QFI may be alternatively calcu-lated by considering purifications | Ψ ϕ i of a given fam-ily of mixed states on an extended Hilbert space ρ ϕ = Tr E {| Ψ ϕ ih Ψ ϕ |} , where by E we denote an ancillary spaceneeded for the purification. It has been proven by Escher et al. (2011) that the QFI of any ρ ϕ is equal to the small- est QFI of its purifications | Ψ ϕ i F Q [ ρ ϕ ] = min Ψ ϕ F Q [ | Ψ ϕ i ] = 4 min Ψ ϕ (cid:26)D ˙Ψ ϕ | ˙Ψ ϕ E − (cid:12)(cid:12)(cid:12)D ˙Ψ ϕ | Ψ ϕ E(cid:12)(cid:12)(cid:12) (cid:27) . (80)Even though minimization over all purification may stillbe challenging, the above formulation may easily be em-ployed in deriving upper bounds on QFI by consideringsome class of purifications. Since upper bounds on QFItranslate to lower bounds on estimation uncertainty thisapproach proved useful in deriving bounds in quantummetrology in presence of decoherence (Escher et al. , 2012,2011).Independently, in (Fujiwara and Imai, 2008) anotherpurification-based QFI definition has been constructed: F Q [ ρ ϕ ] = 4 min Ψ ϕ D ˙Ψ ϕ | ˙Ψ ϕ E . (81)Despite apparent difference, Eqs. (80) and (81) are equiv-alent and one can prove that any purification minimiz-ing one of them is likewise optimal for the other caus-ing the second term of Eq. (80) to vanish. Although forany suboptimal | Ψ ϕ i Eq. (80) must provide a strictlytighter bound on QFI than Eq. (81), the latter defini-tion, owing to its elegant form allows for a direct and ef-ficient procedure for derivation of the precision bounds inquantum metrology (Demkowicz-Dobrzański et al. , 2012;Kołodyński and Demkowicz-Dobrzański, 2013). Deriva-tions of the precision bounds using the above two tech-niques in the context of optical interferometry are dis-cussed in Sec. VIFor completeness, we list below some other importantproperties of the QFI. QFI does not increase under aparameter independent quantum channel F Q ( ρ ϕ ) ≥ F Q [Λ( ρ ϕ )] , (82)where Λ is an arbitrary completely positive (CP) map.QFI appears in the lowest order expansion of the measureof fidelity F between two quantum states F ( ρ , ρ ) = (cid:18) Tr q √ ρ ρ √ ρ (cid:19) , (83) F ( ρ ϕ , ρ ϕ + d ϕ ) = 1 − F Q ( ρ ϕ ) d ϕ + O ( d ϕ ) . (84)QFI is convex F Q X i p i ρ ( i ) ϕ ! ≤ X i p i F Q ( ρ ( i ) ϕ ) , X i p i = 1 , p i ≥ , (85)which reflects the fact that mixing quantum states canonly reduce achievable estimation sensitivity. In a com-monly encountered case, specifically in the context of op-tical interferometry, when the estimated parameter is en-coded on the state by a unitary ρ ϕ = U ϕ ρU † ϕ , U ϕ = e − i ˆ Hϕ , (86)9where ˆ H is the generating “Hamiltonian”, the general for-mula for QFI reads: F Q ( ρ ϕ ) = X i,j |h e i | ˆ H | e j i| ( λ i − λ j ) λ i + λ j (87)where | e i i , λ i form the eigendecomposition of ρ . Notethat in this case the QFI does not depend on the actualvalue of ϕ . For the pure state estimation case, ρ = | ψ ih ψ | ,the QFI is proportional to the variance of ˆ H : F Q ( | ψ ϕ i ) = 4∆ H = 4( h ψ | ˆ H | ψ i − h ψ | ˆ H | ψ i ) (88)and the QCRB (77) takes a particular appealing formresembling the form of the energy-time uncertainty rela-tion: ∆ ϕ ∆ H ≥ N . (89)We conclude the discussion of QFI properties by men-tioning a very recent and elegant result proving that forunitary parameter encodings, QFI is proportional to theconvex roof of the variance of ˆ H (Tóth and Petz, 2013;Yu, 2013): F Q ( ρ ϕ ) = 4 min { p i , | ψ ( i ) i} X i p i ∆ H ( i ) , (90)where the minimum is performed over all decompositionsof ρ = P i p i | ψ ( i ) ih ψ ( i ) | , and ∆ H ( i ) denotes the varianceof ˆ H calculated on | ψ ( i ) i .
2. Bayesian approach
As the quantum mechanical estimation scenario, with aparticular measurement scheme chosen, represents noth-ing but a probabilistic model with outcome probabilities p ϕ ( x ) , we may also apply the Bayesian techniques de-scribed in Sec. IV.A.2. The quantum element of the prob-lem, however, i.e. minimization of the average cost func-tion over the choice of measurements is in general highlynon-trivial. Fortunately, provided the problem possessesa particular kind of symmetry it may be solved using theconcept of covariant measurements (Holevo, 1982).In the context of optical interferometry, it is sufficientto consider the unitary parameter encoding case as de-fined in (86), where the estimated parameter ϕ will cor-respond to the phase difference in an interferometer, seeFig. 5 in Sec. V. Let us denote a general POVM mea-surement as M x and the corresponding estimator as ˜ ϕ .Since we need to minimize the average cost over boththe measurements and the estimators it is convenientto combine these two elements into one by labeling thePOVM elements with the estimated values themselves: M ˜ ϕ = ´ d x M x δ [ ˜ ϕ − ˜ ϕ ( x )] . The expression for the aver-age cost function, (64), takes the form: h C i = ¨ dϕ π d ˜ ϕ π p ( ϕ ) Tr ( U ϕ ρU † ϕ M ˜ ϕ ) C ( ˜ ϕ, ϕ ) . (91) and leaves us with the problem of minimization h C i overa general POVM M ˜ ϕ with standard constraints M ˜ ϕ ≥ , ´ d ˜ ϕ π M ˜ ϕ =
1. Note that for clarity we use the normalizedmeasure d ϕ π .Provided the problem enjoys the ϕ shift symmetry, sothat p ( ϕ + ϕ ) = p ( ϕ ) , C ( ˜ ϕ + ϕ , ϕ + ϕ ) = C ( ˜ ϕ, ϕ ) ,it may be shown that one does not loose optimality byrestricting the class of POVM measurements to M ˜ ϕ = U ˜ ϕ Ξ U † ˜ ϕ , (92)which is a special case of the so-called covariant mea-surements (Bartlett et al. , 2007; Chiribella et al. , 2005;Holevo, 1982). If we take flat prior distribution p ( ϕ ) = 1 ,and the natural cost function C ( ˜ ϕ, ϕ ) = 4 sin [( ˜ ϕ − ϕ ) / introduced in Sec. IV.A.2, symmetry conditions are ful-filled and substituting (92) into (91) we get a simple ex-pression: h C i = Tr (cid:8) h ρ ϕ i C Ξ (cid:9) , (93)where h ρ ϕ i C = 4 ´ dϕ π U ϕ ρU † ϕ sin (cid:0) ϕ (cid:1) is the final quantumstate averaged with the cost function. Looking for theoptimal Bayesian strategy now amounts to minimizationof the above quantity over Ξ with the POVM constraintsrequiring that Ξ ≥ , ´ d ˜ ϕ π U ˜ ϕ Ξ U † ˜ ϕ =
1. As shown inSec. V and Sec. VI this minimization is indeed possiblefor optical interferometric estimation models and allowsto find the optimal Bayesian strategy and the correspond-ing minimal average cost.
V. QUANTUM LIMITS IN DECOHERENCE-FREEINTERFEROMETRY ϕ n a n b | α i| r i | ψ i in ˜ ϕ ( n a , n b ) M x ˜ ϕ ( x ) FIG. 5 Instead of a particular Mach-Zehnder interferometricstrategy a general quantum interferometric scheme involvesa general input probe state | ψ i in that is subject to a unitaryphase delay operation U ϕ followed by a general quantum mea-surement (POVM) M x . Finally, an estimator ˜ ϕ ( x ) is used toobtain the estimated value of the phase delay. In order to analyze the ultimate precision bounds ininterferometry one needs to employ the quantum esti-mation theory introduced in Sec. IV.B. In this sectionwe review the most important results of decoherence-freeinterferometry leaving the analysis of the impact of de-coherence for Sec. VI. This will provide us with precisionbenchmarks to which we will be able to compare realistic0estimation schemes and analyze the reasons for the de-parture of practically achievable precisions from idealizedscenarios.Formally, interferometry may be regarded as a chan-nel estimation problem where a known state | ψ i in is sentthrough a quantum channel U ϕ = e − iϕ ˆ J z , with an un-known parameter ϕ , where ˆ J z = ( a † a − b † b ) is the z component of angular momentum operator introducedin Sec. III.A. Quantum measurement ˆ M x is performedat the output state and the value of ϕ is estimated basedon the obtained outcome x through an estimator ˜ ϕ ( x ) ,see Fig. 5. Pursuing either QFI or Bayesian approachit is possible to derive bounds on achievable precisionthat are valid irrespectively of how sophisticated is themeasurement-estimation strategy employed and how ex-otic the input states of light are. We start by consid-ering definite photon-number states using both QFI andBayesian approaches and then move on to discuss issuesthat arise when discussing fundamental bounds takinginto account states of light with indefinite photon num-ber. A. Quantum Fisher Information approach
As discussed in detail in Sec. IV, the QFI approach isparticularly well suited to analyze problems where onewants to estimate small deviations of ϕ around a knownvalue ϕ , as in this local regime the QCRB, (77), isknown to be saturable. This is, for example, the caseof gravitational-wave interferometry in which one setsthe interferometer at the dark fringe and wants to es-timate small changes in the interference pattern inducedby the passing wave (LIGO Collaboration, 2011, 2013;Pitkin et al. , 2011).Since the state of light at the output, | ψ ϕ i =e − i ϕ ˆ J z | ψ i in , is pure, the QFI may be calculated usingthe simple formula given in Eq. (79). Realizing that | ˙ ψ ϕ i = − i ˆ J z | ψ ϕ i we get the QFI and the correspondingQCRB on the estimation precision: F Q = 4 (cid:16) h ψ ϕ | ˆ J z | ψ ϕ i − |h ψ ϕ | ˆ J z | ψ ϕ i| (cid:17) = 4∆ J z , ∆ ϕ ≥ J z . (94)Note that the form of the QCRB above may be regardedas an analogue of the Heisenberg uncertainty relation forphase and angular momentum.Clearly, according to the above bound, the optimalprobe states for interferometry are the ones that max-imize ∆ J z . We fix the total number of photons andlook for N -photon states maximizing ∆ J z . A General N -photon input state and the explicit form of the corre- sponding QFI read: | ψ i in = N X n =0 c n | n i| N − n i , (95) F Q = 4 N X n =0 | c n | n − N X n =0 | c n | n ! . (96)Let us first consider the situation in which | ψ i in is astate resulting from sending N photons on the balancedbeam-splitter, as discussed in Sec. III.C. This time, how-ever, we do not insist on a particular measurement norestimation scheme, but just want to calculate the cor-responding QCRB on the sensitivity. Written in thephoton-number basis the state takes the form: | ψ i in = N X n =0 s N (cid:18) Nn (cid:19) | n i| N − n i , (97)for which F Q = N and this results in the shot noise boundon precision: ∆ ϕ | N i| i ≥ √ N . (98)Recall, that this bound is saturated with the sim-ple MZ interferometric scheme discussed in Eq. (36),which is a proof that for the considered probe statethis measurement-estimation scheme is optimal. From aparticle-description point of view, see Sec. II.D, the aboveconsidered state is a pure product state with no entan-glement between the photons. More generally, the shotnoise limit, sometimes referred to as the standard quan-tum limit, is valid for all N -photon separable states (Gio-vannetti et al. , 2011; Pezzé and Smerzi, 2009), and goingbeyond this limit requires making use of inter-photon en-tanglement, see Sec. V.D.Let us now investigate general input states, which pos-sibly may be entangled. Consider the state of the form | NOON i = √ ( | N i| i + | i| N i ) which is commonly re-ferred to as the “NOON” state (Bollinger et al. , 1996; Lee et al. , 2002). QFI for such state is given by F Q = N andconsequently ∆ ϕ ≥ N , (99)which is referred to as the
Heisenberg limit . In fact,NOON state gives the best possible precision as it has thebiggest variance of ˆ J z among the states with a given pho-ton number N (Bollinger et al. , 1996; Giovannetti et al. ,2006). Still, the practical usefulness of the NOON statesis doubtful. The difficulty in their preparation increasesdramatically with increasing N , and with present tech-nology the experiments are limited to relatively small N ,e.g. N = 4 (Nagata et al. , 2007) or N = 5 (Afek et al. ,12010). Moreover, even if prepared, their extreme suscep-tibility to decoherence with increasing N , see Sec. VI,makes them hardly useful in any realistic scenario unless N is restricted to small values. Taking into account thatexperimentally accessible squeezed states of light offera comparable performance in the decoherence-free sce-nario, see Sec. V.C, and basically optimal asymptoticperformance in the presence of decoherence, see Sec. VI,there is not much in favor of the N00N states apart fromtheir conceptual appeal. B. Bayesian approach
Let us now look for the fundamental precision boundsin the Bayesian approach (Berry and Wiseman, 2000;Hradil et al. , 1996). We assume the flat prior distribu-tion p ( ϕ ) = 1 / π reflecting our complete initial ignoranceon the true phase value, and the natural cost function C ( ˜ ϕ, ϕ ) = 4 sin [( ˜ ϕ − ϕ ) / , see Sec. IV.A.2. Thanks tothe phase shift symmetry of the problem, see Sec. IV.B.2,we can restrict the class of measurements to covari-ant measurements M ˜ ϕ = U ˜ ϕ Ξ U † ˜ ϕ , where Ξ is the seedmeasurement operator, to be optimized below. UsingEq. (93), and noting that ´ π dϕ π ( ϕ/ iϕ ( n − m ) =2 δ nm − ( δ n,m − + δ n,m +1 ) , the averaged cost for a general N -photon input state | ψ i = P Nn =0 c n | n, N − n i reads: h C i = 2 − Re N X n =1 c ∗ n c n − Ξ n,n − ! . (100)Because Ξ is Hermitian, the completeness condition ´ dϕ π ˆ U ϕ Ξ ˆ U † ϕ = Ξ nn = 1 , while due tothe positive semi-definiteness condition Ξ ≥ , | Ξ nm | ≤√ Ξ nn Ξ mm = 1 . Therefore, The real part in the sub-tracted term in Eq. (100) can at most be P Nn =1 | c n || c n − | ,which will be the case for Ξ n,m = e i( ξ n − ξ m ) , where ξ n = arg( c n ) . This is a legitimate positive semi-definiteoperator, as it can be written as Ξ = | e N ih e N | , with | e N i = P Nn =0 e i ξ n | n, N − n i (Chiribella et al. , 2005;Holevo, 1982). Thus, for a given input state the opti-mal Bayesian measurement-estimation strategy yields h C i = 2 − N X n =1 | c n c n − | ! . (101)For the uncorrelated input state (97), the average costreads: h C i = 2 − N N − X n =0 s(cid:18) Nn (cid:19)(cid:18) Nn + 1 (cid:19)! N →∞ ≈ N . (102)Since in the limit of small estimation uncertainty the con-sidered cost function approaches the MSE, we may con-clude that in the limit of large N : ∆ ϕ N →∞ ≈ √ N , (103) which coincides with the standard shot-noise limit de-rived within the QFI approach.Note a subtle difference between the above solutionand the solution of the optimal Bayesian transmissioncoefficient estimation problem discussed in Sec. IV.A.3with the ϕ parametrization employed. The formulas forprobabilities in Sec. IV.A.3, can be regarded as arisingfrom measuring each of uncorrelated photons leaving theinterferometer independently, while in the present con-siderations we have allowed for arbitrary quantum mea-surements, which are in general collective. Importantly,we account for the adaptive protocols in which a mea-surement on a subsequent photon depends on the re-sults obtained previously (Kołodyński and Demkowicz-Dobrzański, 2010)—practically these are usually addi-tional controlled phase shifts allowing to keep the setupat the optimal operation point (Higgins et al. , 2007).This approach is therefore more general and in partic-ular, does not suffer from a ± ϕ ambiguity that forcedus to restrict the estimated region to [0 , π ) in order toobtain nontrivial results given in Eq. (75).Let us now look for the optimal input states. FromEq. (102) it is clear that we may restrict ourselves toreal c n . Denoting by c the vector containing the statecoefficients c n , we rewrite the formula for h C i in a moreappealing form h C i = 2 − c T Ac , A n,n − = A n − ,n = 1 , (104)from which it is clear that minimizing the cost function isequivalent to finding the eigenvector with maximal eigen-value of the matrix A , which has all its entries zero ex-cept for its first off-diagonals. This can be done analyti-cally (Berry and Wiseman, 2000; Luis and Perina, 1996;Summy and Pegg, 1990) and the optimal state, which wewill refer to as the sine state, together with the resultingcost read | ψ i in = N X n =0 r
22 + N sin (cid:16) n + 1 N + 2 π (cid:17) | n i| N − n i , (105) h C i = 2 (cid:20) − cos (cid:18) πN + 2 (cid:19)(cid:21) N →∞ ≈ π N . (106)Again, in the large N limit we may identify the aver-age cost with the average MSE, so that the asymptoticprecision reads: ∆ ϕ N →∞ ≈ πN . (107)Analogously, as in the QFI approach we arrive at the /N Heisenberg scaling of precision, but with an addi-tional constant factor π , reflecting the fact that Bayesianapproach is more demanding as it requires the strategyto work well under complete prior ignorance of the valueof the estimated phase. Note also that the structure ofthe optimal states is completely different from the NOON2states. In fact, the NOON states are useless in absenceof any prior knowledge on the phase, since they are in-variant under π/N phase shift, and hence cannot unam-biguously resolve phases differing by this amount. C. Indefinite photon-number states
We now consider a more general class of states withindefinite photon numbers and look for optimal probestates treating the average photon number h N i as a fixedresource. A state with an indefinite number of photonsmay posses in general coherences between sectors withdifferent total numbers of particles. These coherencesmay in principle improve estimation precision. However,a photon number measurement performed at the out-put ports projects the state on one of the sectors andnecessarily all coherences between different total photonnumber sectors are destroyed. In order to benefit fromthese coherences, one needs to make use of a more generalscheme such as e.g. homodyne detection, where an ad-ditional phase reference beam is needed, typically calledthe local oscillator, which is being mixed with the signallight at the detection stage. Usually, the local oscilla-tor is assumed to be strong, classical field with a welldefined phase. In other words, it provides one with ref-erence frame with respect to which phase of the signalbeams can be measured (Bartlett et al. , 2007; Mølmer,1997). Thus, it is crucial to explicitly state whether thereference beam is included in the overall energy budget oris it treated as a free resource. Otherwise one my arriveat conflicting statements on the achievable fundamentalbounds (Jarzyna and Demkowicz-Dobrzański, 2012).
1. Role of the reference beam
As an illustrative example, consider an artificial onemode scheme with input in coherent state | ψ i = | α i which passes through the phase delay ϕ . Strictly speak-ing, this is not an interferometer and one may wonderhow one can possibly get any information on the phaseby measuring the output state. Still | ψ i evolves into aformally different state | ψ ϕ i = U ϕ | α i = | α e − i ϕ i , where U ϕ = e − i ϕ ˆ n , and since the corresponding QFI is non-zero F Q = 4 (cid:0) h α | ˆ n | α i − |h α | ˆ n | α i| (cid:1) = 4 | α | , (108)it is in principle possible to draw some information onthe phase by measuring | ψ ϕ i . Clearly, the measure-ment required cannot be a direct photon-number mea-surement, and an additional phase reference beam needsto be mixed with the state before sending the light to thedetectors. In a fair approach one should include the ref-erence beam into the setup and assume that whole stateof signal+reference beam is averaged over a global un-defined phase. This formalizes the notion of the relative phase delay - it is defined with respect to reference beamand there is no such thing like absolute phase delay.More formally, let | ψ i ar = | α i a | β i r , be the original co-herent state used for sensing the phase, accompanied bya coherent reference beam | β i . The corresponding out-put state reads | ψ ϕ i ar = | α e − i ϕ i a | β i r . Now, the phase ϕ plays the role of the relative phase shift between the twomodes, with a clear physical interpretation. The com-bined phase shift in the two modes, i.e. an operation U aθ U rθ = e − i θ (ˆ n a +ˆ n r ) has no physical significance as it isnot detectable without an . . . additional reference beam.Hence, before calculating the QFI or any other quantitydetermining fundamental precision bounds, one shouldfirst average the state | ψ i ar over the combined phase shiftand treat the resulting density matrix as the input probestate ρ = ˆ π dθ π U aθ U rθ | ψ i ar h ψ | ⊗ | β ih β | U a † θ U r † θ = ∞ X N =0 p N ρ N . (109)This operation destroys all the coherences between sec-tors with different total photon number, and the resultingstate is a mixture of states ρ N with different total photonnumbers N , appearing with probabilities p N .In the absence of a reference beam, i.e. when β = 0 , theabove averaging kills all the coherences between termswith different photon numbers in the mode a : ρ = ˆ dθ π | α e − iθ ih α e − iθ | ⊗ | ih | == e −| α | ∞ X n =0 | α | n n ! | n ih n | ⊗ | ih | (110)which results in a state insensitive to phase delays andgives F = 0 , restoring our physical intuition. On theother hand, F = 4 | α | obtained previously is recoveredin the limit | β | → ∞ , meaning that reference beam isclassical—consists of many more photons than the signalbeams.The above averaging prescription, is valid in generalalso when label a refers to more than one mode. Consid-ering the standard Mach-Zehnder interferometer fed witha state | ψ i ab with an indefinite photon numbers and noadditional reference beam available one again needs toperform the averaging over a common phase shift. If thisis not done, one may obtain conflicting results on e.g.QFI for seemingly equivalent phase shift operations suchas U ϕ = e − iˆ n a ϕ or U ′ ϕ = e − i(ˆ n a − ˆ n b ) ϕ/ . The reason isthat, without the common phase averaging, one implic-itly assumes the existence of a strong external classicalphase reference to with respect to which the phase shiftsare defined. In particular, U ϕ assumes that the secondmode is perfectly locked with the external reference beamand is not affected by the phase shift, whereas U ′ ϕ as-sumes that there are exactly opposite phase shifts in thetwo modes with respect to the reference. Different choices3of „phase-shift distribution” may lead to a factor of oreven factor of discrepancies in the reported QFIs in ap-parently equivalent optical phase estimation schemes—see (Jarzyna and Demkowicz-Dobrzański, 2012) for fur-ther discussion and compare with some of the resultsthat were obtained without the averaging performed (Joo et al. , 2011; Spagnolo et al. , 2012).One can also understand why ignoring the need fora reference beam may result in underestimating the re-quired energy resources. Consider a singe mode statewith an indefinite photon number | ψ i = P Nn =0 c n | n i evolving under e − i ϕ ˆ n and note that, from the phase sens-ing point of view, this situation is formally equivalentto a two-mode state with a definite photon number N : | ψ i ab = P Nn =0 c n | n i| N − n i , evolving under e − i ϕ ˆ n a . Still,the average photon number consumed in the one modecase is h N i = P Nn =0 | c n | n and is in general smaller than N .
2. Optimal indefinite photon number strategies
Looking for the optimal states with fixed average pho-ton number is in general more difficult than in the definitephoton-number case. Still, if we agree with the above-advocated approach to average all the input states over acommon phase-shift transformation as in Eq. (109), thenthe resulting state is a probabilistic mixture of definitephoton number states. Intuitively, it is then clear that,instead of sending the considered averaged state, it ismore advantageous to have information which particularcomponent ρ N of the mixture is being sent. This wouldallow to adjust the measurement-estimation procedure toa given component and improve the overall performance.This intuition is reflected by the properties of both theQFI and the Bayesian cost, which are respectively convexand concave quantities (Helstrom, 1976): F Q X N p N ρ N ! ≤ X N p N F Q ( ρ N ) , (111) * C X N p N ρ N !+ ≥ X N p N h C ( ρ N ) i . (112)This, however, implies that knowing the solution for theoptimal definite photon number probe states, by adjust-ing the probabilities p N with which different optimal ρ N are being sent, we may determine the optimal strategieswith the average photon-number fixed.Taking the QFI approach for a moment, we recall thatthe optimal N -photon state, the NOON state, yields F Q ( ρ N ) = N . Let us consider a strategy where a vac-uum state and the NOON state are sent with probabili-ties − p and p respectively, with the constraint on theaverage photon number pN = h N i . The correspondingQFI reads F Q = (1 − p ) · pN . Substituting p = h N i /N we get F Q = N h N i . Therefore, while keeping h N i fixedwe may increase N arbitrarily and in principle reach F Q = ∞ , suggesting the possibility of arbitrary goodsensing precisions (Rivas and Luis, 2012; Zhang et al. ,2013). Note in particular, that a naive generalizationof the Heisenberg limit to ∆ ϕ ≥ h N i , does not hold,and the strategies beating this bound are referred to assub-Heisenberg strategies (Anisimov et al. , 2010). A uni-versally valid bound may be written as ∆ ϕ ≥ / q h ˆ N i (Hofmann, 2009), but the question remains, whether thebound is saturable, and in particular does quantum me-chanics indeed allows for practically useful estimationprotocols leading to the sub-Heisenberg scaling of pre-cision. Closer investigations of that problem proves suchhypothesis to be false (Berry et al. , 2012; Giovannetti andMaccone, 2012; Tsang, 2012; Zwierz et al. , 2010). In prin-ciple we may achieve the sub-Heisenberg precision in thelocal estimation regime but in order for the local strat-egy to be valid, we should know the value of estimatedparameter with prior precision of the same order as theone we want to obtain, what makes the utility of the pro-cedure questionable. Actually, if no such assumption onthe priori knowledge is made, the Heisenberg scaling inthe form ∆ ϕ = const / h N i is the best possible scaling ofprecision.This claim can also be confirmed within the Bayesianapproach with flat prior phase distribution. For large N the minimal Bayesian cost behaves like h C ( ρ N ) i = π /N , see Eq. (105). Since this function is convex, tak-ing convex combinations of the cost for two different totalphoton numbers N , N , such that p N + p N = h N i will yield the cost higher than π / h N i , and the corre-sponding uncertainty ∆ ϕ ≥ π/ h N i , indicating the uni-versal validity of the Heisenberg scaling of precision.
3. Gaussian states
From a practical point of view, rather than looking forthe optimal indefinite photon number states for interfer-ometry it is more important to analyze experimentallyaccessible Gaussian states. The paradigmatic example ofa Gaussian state applied in quantum enhanced interfer-ometry is the two mode state | α i| r i —coherent state inmode a and squeezed vacuum in mode b . We have alreadydiscussed this example in Sec. III.D, and calculated theprecision for a simple measurement-estimation scheme.For such states, sent through a fifty-fifty beam splitter,quantum Fisher information can been calculated explic-itly (Jarzyna and Demkowicz-Dobrzański, 2012; Ono andHofmann, 2010; Pezzé and Smerzi, 2008): F Q = | α | e r + sinh r. (113)For the extreme cases | α | = 0 , sinh r = h N i and | α | = h N i , sinh r = 0 this formula gives F Q = h N i ,4implying the shot noise scaling. Most importantly, op-timization of Eq. (113) over α and r with constraint | α | + sinh r = h N i gives asymptotically the Heisen-berg limit ∆ ϕ ∼ / h N i , making this strategy as goodas the NOON -one for large number of photons. More-over, this bound on precision can be saturated by esti-mation strategies based on photon-number (Pezzé andSmerzi, 2008; Seshadreesan et al. , 2011) or homodyne(D’Ariano et al. , 1995) measurements. This also provesthat a simple measurement-estimation strategy discussedin Sec. III.D which yielded / h N i / scaling of precisionis not optimal. Unlike the simple interferometric schemewhere it was optimal to dedicate approximately p h N i photons to the squeezed beam, from the QFI point ofview it is optimal to equally divide the number of pho-tons between the coherent and the squeezed beam.More generally, finding the fundamental limit on pre-cision achievable with general Gaussian states, requiresoptimization of the QFI or the Bayesian average costfunction over general two-mode Gaussian input states,specified by the covariance matrix and the vector of firstmoments, see Sec. II.B. For the decoherence-free case thiswas done in Pinel et al. (2012, 2013). Crucial observationis that for pure states, the overlap between two M-modeGaussian states | ψ ϕ i and | ψ ϕ + dϕ i is given by (up to thesecond order in d ϕ ) |h ψ ϕ | ψ ϕ + dϕ i| = 1 − d ϕ π ) ˆ (cid:18) d W ( z ) dϕ (cid:19) d M z ! (114)where W ( x ) is the Wigner function (5) of state | ψ ϕ i .Thus, because |h ψ ϕ | ψ ϕ + dϕ i| = 1 − F Q d ϕ we may writethat ∆ ϕ ≥ π ) ˆ (cid:18) d W ( z ) d ϕ (cid:19) d M z ! − . (115)In terms of the covariance matrix σ and the first moments h z i the formula takes an explicit form: ∆ ϕ ≥ d h z i dϕ T σ − d h z i dϕ + 14 Tr (cid:18) dσdϕ σ − (cid:19) !! − . (116)Formal optimization of the above equation was done byPinel et al. (2012), however, the result was a one-modesqueezed-vacuum state, which in order to carry phase in-formation needs to be assisted by a reference beam. Un-fortunately, performing a common phase-averaging pro-cedure described in Sec. V.C.1 in order to calculate theprecision in the absence of additional phase reference de-stroys the Gaussian structure of the state and makes theoptimization intractable. Luckily, for path symmetricstates, i.e. the states invariant under the exchange ofinterferometer arms, the phase averaging procedure doesnot affect the QFI (Jarzyna and Demkowicz-Dobrzański, 2012). Hence, assuming the path-symmetry the optimalGaussian state is given by | r i| r i with sinh r = h N i / —two squeezed vacuums send into the input ports of theinterferometer—and its corresponding QFI leads to theQCRB ∆ ϕ ≥ p h N i ( h N i + 2) ≈ h N i (117)The state also achieves the Heisenberg limit for a largenumber of photons in the setup but it does not requireany external phase reference. It is also worth notingthat this state, while being mode-separable is particle-entangled and is feasible to prepare with current tech-nology for moderate squeezing strengths. However, theenhancement over optimal squeezed-coherent strategy israther small and vanish for large number of photons.Precisions obtained in squeezed-coherent and squeezed-squeezed scenarios are depicted in Fig. 6. h N i ∆ ϕ ≤ h N i ∆ ϕ ≥ h N i FIG. 6 Limits on precision obtained within QFI approachwhen using two optimally squeezed states in both modes | r i| r i (black, solid), coherent and squeezed-vacuum states | α i| r i (black, dashed). For comparison precision achievable withsimple coherent and squeezed vacuum MZ interferometricscheme discussed in Sec. III.D is also depicted (gray, dashed). One can also study Gaussian states within theBayesian framework. Optimal seed operator can be eas-ily generalized from the definite photon number case to
Ξ = P ∞ N =0 | e N ih e N | . Conceptually, the whole treatmentis the same as in the definite photon number case. How-ever, the expressions and calculations are very involvedand will not be presented here. D. Role of entanglement
The issue of entanglement is crucial in quantum in-terferometry as it is known that only entangled statescan beat the shot noise scaling (Pezzé and Smerzi, 2009).This statement is sometimes questioned, pointing outthe example of the squeezed+coherent light strategy,where the interferometer is fed with seemingly unentan-gled | α i| r i input state. The reason of confusion is the5conflict of notions of mode and particle entanglement.As discussed in detail in Sec. II.E, the two notions arenot compatible, and there are states which are particleentangled, while having no mode entanglement and viceversa. In the context of interferometry it is the particle entanglement that is the source of quantum enhancementof precision. In order to avoid criticism based on theground of fundamental indistinguishability of particlesand therefore a questionable physical content of the dis-tinguishable particle-based entanglement picture on thefundamental level (Benatti et al. , 2010), we should stressthat when considering models involving indistinguishableparticles one should regard this statement as a formal(but still a meaningful and useful) criterion where theparticles are treated as formally distinguishable as de-scribed in Sec. II.D.To see this, let us consider first a separable input stateof N photons of the form ρ = ρ ⊗ · · · ⊗ ρ N , where ρ i denotes the state of the i -th photon. Since the phaseshift evolution affects each of the photons independently ρ ϕ = U ⊗ Nϕ ρU ⊗ N † ϕ and the QFI is additive on productstates we may write: F Q ( ρ ) = N X i =1 F Q ( ρ i ) ≤ N F Q ( ρ max ) (118)where ρ max denotes state from the set { ρ i } i =1 ,...,N forwhich QFI takes the largest value. But for a one photonstate, the maximum value of QFI is equal to , so F ( ρ ) ≤ N, ∆ ϕ ≥ √ N . (119)For general separable states ρ = P i p i ρ ( i )1 ⊗ · · · ⊗ ρ ( i ) N itis sufficient to use the convexity of QFI, together withEq. (119) to obtain the same conclusion. Above resultsimply that QFI, or precision, can be interpreted as aparticle-entanglement witness, i.e. all states that give pre-cision scaling better than the shot noise must be particle-entangled (Hyllus et al. , 2012; Tóth, 2012).The seemingly unentangled state | α i| r i when pro-jected on the definite total photon number sector, indeedcontains particle entanglement as was demonstrated inSec. II.E. This fact should be viewed as the fundamentalsource of its ability for performing quantum-enhancedsensing. It is also worth stressing, that unlike modeentanglement, particle entanglement is invariant underpassive optical transformation like beam splitters, delaylines and mirrors, which makes it a sensible quantity tobe treated as a resource for quantum enhanced interfer-ometry. E. Multi-pass protocols
A common method, used in e.g. gravitational wavedetectors, to improve interferometric precision is to let
FIG. 7 A multi-pass interferometric protocol. A standardphase delay element is replaced by a setup which makes thebeam to pass through the phase delay multiple number oftimes. the light bounce back and forth through the phase de-lay element many times so that the phase delay signalis enhanced as shown in Fig. 7. This method is used inGEO600 experiment (LIGO Collaboration, 2011), wherethe light bounces twice in each of the sensing arms, mak-ing the detector as sensitive as the one with arms twiceas long. Up to some approximation, one can also viewthe Fabry-Perrot cavities placed on top of the Mach-Zehnder design as devices forcing each of the photon topass multiple-times through the arms of the interferome-ter and acquire a multiple of the phase delay (Berry et al. ,2009; Demkowicz-Dobrzański et al. , 2013).Consider a single photon in the state (after the firstbeam splitter) | ψ i in = √ ( | i + | i ) . After passingthrough the phase shift N times it evolves into | ψ ϕ i = √ ( | i + e − i Nϕ | i ) . The phase is acquired N timesfaster compared with a single pass case, mimicking thebehavior of a single pass experiment with a NOON state.Hence, the precision may in principle be improved by afactor of N . Treating the number of single photon passesas a resource, it has been demonstrated experimentally(Higgins et al. , 2007) that in the absence of noise sucha device can indeed achieve the Heisenberg scaling with-out resorting to entanglement and efficiency of variousmulti-pass protocols has been analyzed in detail in (Berry et al. , 2009). This is not to say, that all quantum strate-gies are formally equivalent to single-photon multi-passstrategies. As will be discussed in the next section, theNOON states are extremely susceptible to decoherence,in particular loss, and this property is shared by themulti-pass strategies. Other quantum strategies provemore advantageous in this case, and they do not havea simple multi-pass equivalent (Demkowicz-Dobrzański,2010; Kaftal and Demkowicz-Dobrzanski, 2014). VI. QUANTUM LIMITS IN REALISTICINTERFEROMETRY
In this section we revisit the ultimate limits on preci-sion derived in Sec. V taking into account realistic noiseeffects. We study three decoherence processes that aretypically taken into account when discussing imperfec-6tions in interferometric setups. We consider the effectsof phase diffusion , photonic losses and the impact of im-perfect visibility , see Fig. 8. In order to establish the ul-timate limits on the estimation performance, we first an-alyze the above three decoherence models using the QFIperspective and secondly compare the bounds obtainedwith the ones derived within the Bayesian approach.For the most part of this section, we will consider inputstates with definite number of photons, N , so that ρ in = | ψ i in h ψ | , | ψ i in = N X n =0 c n | n i| N − n i . (120)Similarly as in Sec. V, this will again be sufficient todraw conclusions also on the performance of indefinitephoton number states, which will be discussed in detailin Sec. VI.D.1.In what follows it will sometimes prove useful to switchfrom mode to particle description, see Sec. II.D, andtreat photons formally as distinguishable particles butprepared in a symmetrized state. This approach is il-lustrated in Fig. 9 where each photon is represented bya different horizontal line, and travels through a phaseencoding transformation U ( i ) ϕ = e − i ϕ ˆ σ ( i ) z / , where ˆ σ ( i ) z isa z Pauli matrix acting on the i -th qubit. The com-bined phase encoding operation is a simple tensor prod-uct U ⊗ Nϕ = e − i ϕ P i ˆ σ ( i ) z = e − i ϕ ˆ J z , recovering the familiarformula but with ˆ J z being now interpreted as the z com-ponent of the total angular momentum which is the sumof individual angular momenta. The photons are thensubject to decoherence that acts in either correlated oruncorrelated manner. In the case of phase diffusion (i)the decoherence has a collective character since each ofthe photons experiences the same fluctuation of the phasebeing sensed, while in the case of loss (ii) and imperfectvisibility (iii) the decoherence map has a tensor structure Λ ⊗ N reflecting the fact that it affects each photon inde-pendently. In the latter case of independent decoherencemodels the overall state evolution is uncorrelated andmay be written as: ρ ϕ = Λ ⊗ Nϕ ( ρ in ) , Λ ϕ ( · ) = Λ( U ϕ · U † ϕ ) . (121) A. Decoherence models
In general, decoherence is a consequence of the uncon-trolled interactions of a quantum system with the en-vironment. Provided the system is initially decoupledfrom the environment, the general evolution of a quan-tum system interacting with the environment mathemat-ically corresponds to a completely positive trace preserv-ing map Λ . Every Λ can be written using the Krausrepresentation (Nielsen and Chuang, 2000): ρ out = Λ( ρ in ) = X i K i ρ in K † i , X i K † i K i = , (122) (i) (ii) (iii) m ea s u r e m e n t ab | ψ i in η a η b FIG. 8 Schematic description of the decoherence processesdiscussed that affect the performance of an optical interferom-eter: (i) phase diffusion representing stochastic fluctuationsof the estimated phase delay, (ii) losses in the respective a / b arms represented by fictitious beam splitters with ≤ η a/b ≤ transmission coefficients, (iii) imperfect visibility indicated bya mode mismatch of the beams interfering at the output beamsplitter ΛΛΛ ρ ϕ (ii), (iii) U ϕ U ϕ U ϕ Λ (i) | ψ i in FIG. 9 General metrological scheme in case of photons beingtreated as formally distinguishable particles. Each photontravels through a phase encoding transformation U ϕ . Apartfrom that all photons are subject to either correlated (i)(phase diffusion), or uncorrelated (ii), (iii) (loss, imperfectvisibility) decoherence process. where K i are called the Kraus operators.Effects of decoherence inside an interferometer aretaken into account by replacing the unitary transforma-tion U ϕ describing the action of the ideal interferometer,see Sec. V, with its noisy variant Λ ϕ : ρ ϕ = Λ ϕ ( ρ in ) = X i K i U ϕ ρ in U † ϕ K † i ! . (123)The formal structure of the above formula corresponds toa situation in which decoherence happens after the uni-tary phase encoding. This of course might not be true ingeneral. Still, for all the models considered in this reviewthe decoherence part and the unitary part commute andtherefore the order in which they are written is a matterof convenience.7
1. Phase diffusion
Phase diffusion , also termed as the collective dephasing or the phase noise represents the effect of fluctuation ofthe estimated phase delay ϕ . Such effect may be causedby any process that stochastically varies the effective op-tical lengths traveled by the photons, such as thermal de-formations or the micro-motions of the optical elements.We model the optical interferometry in the presence ofphase diffusion process by the following map: ρ ϕ = Λ ϕ ( ρ in ) = ∞ ˆ −∞ d φ p ϕ ( φ ) U φ ρ in U † φ (124)where the phase delay is a random variable φ distributedaccording to probability distribution p ϕ ( φ ) . Note thatthe above form is actually the Kraus representation ofthe map Λ ϕ with Kraus operators K φ = p p ϕ ( φ ) U φ . Incase p ϕ ( φ ) is a Gaussian distribution with variance Γ andthe mean equal to the estimated parameter ϕ , p ϕ ( φ ) = √ π Γ e − ( φ − ϕ )22Γ , the output state reads explicitly (Genoni et al. , 2011): ρ ϕ = N X n,m =0 c n c ⋆m e − Γ2 ( n − m ) e − i ( n − m ) ϕ | n, N − n ih m, N − m | , (125)where c n are parameters of the input state given as in(120). The above equation indicates that due to thephase diffusion the off-diagonal elements of ρ ϕ are ex-ponentially suppressed at a rate increasing in the anti-diagonal directions.
2. Photonic losses
In the lossy interferometer scenario, the fictitiousbeam-splitters introduced in the interferometer armswith respective power transmission coefficient η a/b ac-count for the probability of photons to leak out. Such aloss model is relatively general, as due to the commutativ-ity of the noise with the phase accumulation (Demkowicz-Dobrzanski et al. , 2009), it accounts for the photoniclosses happening at any stage of the phase sensing pro-cess. Moreover, losses at the detection as well as thepreparation stages can be moved inside the interferom-eter provided they are equal in both arms. This makesthe model applicable in typical experimental realizationof quantum enhanced interferometry (Kacprowicz et al. ,2010; Spagnolo et al. , 2012; Vitelli et al. , 2010), andmost notably, when analyzing bounds on quantum en-hancement in gravitational-wave detectors (Demkowicz-Dobrzański et al. , 2013).Loss decoherence map Λ may be formally describedusing the following set of Kraus operators (Dorner et al. , 2009): K l a ,l b = s (1 − η a ) l a l a ! η ˆ a † ˆ aa ˆ a l a s (1 − η b ) l b l b ! η ˆ b † ˆ bb ˆ b l b (126)where the values of index l a/b corresponds to the numberof photons lost in mode a/b respectively. For a general N -photon input state of the form (120), the density matrixrepresenting the output state of the lossy interferometerreads ρ ϕ = Λ( U ϕ ρ in U † ϕ ) == N M N ′ =0 N − N ′ X l a =0 ( lb = N − N ′− la ) p l a ,l b | ξ l a ,l b ( ϕ ) ih ξ l a ,l b ( ϕ ) | , (127)where p l a ,l b == P Nn =0 | c n | b ( l a ,l b ) n is the binomially dis-tributed probability of losing l a and l b photons in armsa and b respectively, with b ( l a ,l b ) n = (cid:18) nl a (cid:19) η n − l a a (1 − η a ) l a (cid:18) N − nl b (cid:19) η N − n − l b b (1 − η b ) l b , (128)while the corresponding conditional pure states read: | ξ l a ,l b ( ϕ ) i = N − l b X n = l a c n e − i nϕ √ p l a ,l b q b ( l a ,l b ) n | n − l a , N − n − l b i . (129)The direct sum in Eq. (127) indicates that the outputstates of different total number of surviving photons, N ′ ,belong to orthogonal subspaces, which in principle couldbe distinguished by a non-demolition, photon-numbercounting measurement.In the particle-approach when photons are consideredas formally distinguishable particles, the loss process actson each of the photons independently, see Fig. 9, sothat the overall decoherence process has a tensor productstructure Λ ⊗ N , with Λ being a single particle loss trans-formation. At the input stage, each photon occupies atwo-dimensional Hilbert space spanned by vectors | a i , | b i representing the photon traveling in the mode a/b respec-tively. In order to describe loss, however, and formallykeep the number of particles constant, one needs to intro-duce a third photonic state at the output, | vac i , repre-senting the state of the photon being lost. Then formally, Λ maps states from the input two-dimensional Hilbertspace to the output three-dimensional Hilbert space, andcan be fully specified by means of the Kraus representa-tion, Λ( ρ ) = P i =1 K i ρK † i , where K , K , K are givenrespectively by the following matrices: √ η a √ η b , √ − η a , √ − η b . (130)Intuitively, the above Kraus operators account for nophoton loss, photon loss in mode a and photon loss in8mode b respectively. When applied to symmetrized in-put states, this loss model yields output states ρ ϕ = Λ ⊗ N ( U ⊗ Nϕ ρ in U †⊗ Nϕ ) = Λ ⊗ Nϕ ( ρ in ) , (131)equivalent to the ones given in Eq. (127), where U ϕ is asingle photon phase shift operation U ϕ = e − i ϕ ˆ σ z / .
3. Imperfect visibility
In real-life optical interferometric experiments, it is al-ways the case that the light beams employed do not con-tribute completely to the interference pattern. Due tospatiotemporal or polarization mode-mismatch, causedfor example by imperfect wave-packet preparation ormisalignment in the optical elements, the visibility ofthe interference pattern is diminished (Leonhardt, 1997;Loudon, 2000). This effect may be formally described asan effective loss of coherence between the two arms a and b of an interferometer.Consider a single photon in a superposition state ofbeing in modes a and b respectively: | ψ i = α | a i + β | b i .If other degrees of freedom such as e.g. polarization,temporal profile etc. were identical for the two states | a i , | b i , we could formally write ( α | a i + β | b i ) | i X , where | i X denotes the common state of additional degrees offreedom. Loss of coherence may be formally described asthe transformation of the state | ψ i| i X into | Ψ i = α (cid:16) √ η | a i| i X + p − η | a i| + i X (cid:17) ++ β (cid:16) √ η | b i| i X + p − η | b i|−i X (cid:17) , (132)where | + i X , |−i X are states orthogonal to | i X , corre-sponding to photon traveling in e.g. orthogonal transver-sal spacial modes as depicted in Fig. 8 (iii), in which caseparameter η can be interpreted as transmission of ficti-tious beam splitters that split the light into two orthog-onal modes. Assuming we do not control the additionaldegrees of freedom the effective state of the photon is ob-tained by tracing out the above state over X , obtainingthe effective single-photon decoherence map: Λ( | ψ ih ψ | ) = Tr X ( | Ψ ih Ψ | ) = (cid:18) | α | αβ ∗ ηα ∗ βη | β | (cid:19) , (133)where the off-diagonal terms responsible for coherence,are attenuated by coefficient η , what corresponds to thestandard dephasing map (Nielsen and Chuang, 2000).Written using the Kraus representation, the above mapreads: Λ( ρ ) = X i =1 K i ρK † i , K = r η , K = r − η σ z . (134) Note that similarly to the loss model we have modeledthe noise with the use of fictitious beam-splitters to vi-sualize the effects of decoherence. Now, as we know thata beam-splitter acts on the photons contained in its two-mode input state in an uncorrelated manner, the effectivemap on the full N -photon input state is Λ ⊗ N . In case ofatomic systems, this would be a typical local dephasingmodel describing uncorrelated loss of coherence betweenthe two relevant atomic levels (Huelga et al. , 1997). Still,there is a substantial difference from the loss models asthe dephased photons are assumed to remain within thespatially confined beams of the interferometer arms.We can relate the two models by a simple observation,namely that if the photons lost in the loss model with η a = η b = η were incoherently injected back into thearms of the interferometer, we would recover the localdephasing model with the corresponding parameter η .It should therefore come as no surprise, when we derivebounds on precisions for the two models in Sec. VI.B.1and Sec. VI.B.2, that for the same η the local dephas-ing (imperfect visibility) model implies more stringentbounds on achievable precision than the loss model. In-tuitively, it is better to get rid of the photons that losttheir coherence and do not carry information about thephase, rather than to inject them back into the setup.The structure of the output state ρ ϕ is more complexthan in the case of phase diffusion and loss models. Thisis because the local dephasing noise not only transformsthe input state into a mixed state, but due to tracing outsome degrees of freedom, the output state ρ ϕ = Λ ⊗ Nϕ ( ρ in ) == X i ,...,i N =0 K i ⊗· · ·⊗ K i N U ⊗ Nϕ ρ in U ⊗ N † ϕ K † i ⊗· · ·⊗ K † i N (135)is no longer supported on the bosonic space spanned bythe fully symmetric states | n i| N − n i . This makes it im-possible to use the mode-description in characterizationof the process. Even though it is possible to write downthe explicit form of the above state (Fröwis et al. , 2014;Jarzyna and Demkowicz-Dobrzanski, 2014) decomposingthe state into SU(2) irreducible subspaces, we will notpresent it here for the sake of conciseness, especially thatit will not be needed in derivation of the fundamentalbounds. B. Bounds in the QFI approach
Once we have the formula for the output states ρ ϕ , given a particular decoherence model, we may useEq. (77) to calculate QFI, F Q ( ρ ϕ ) , which sets the limit onpractically achievable precision of estimation of ϕ . In or-der to obtain the fundamental precision bound for a given9number N of photons used, we need to maximize the re-sulting F Q over input states | ψ i in , which will in general bevery different from the NOON states which maximize theQFI in the decoherence-free case. This is due to the factthat the NOON states are extremely susceptible to deco-herence, as loss of e.g. a single photon makes the statecompletely useless for phase sensing. Unfortunately, formixed states, the computation of the QFI requires in gen-eral performing the eigenvalue decomposition of ρ ϕ andsuch a minimization ceases to be effective for large N .Therefore, while it is relatively easy to obtain numeri-cal bounds on precision and the form of optimal statesfor moderate N (Demkowicz-Dobrzański, 2010; Dorner et al. , 2009; Huelga et al. , 1997), going to the large N regime poses a huge numerical challenge, making deter-mination of the asymptotic bounds for N → ∞ withbrute force optimization methods infeasible.Over the past few years, elegant methods have beenproposed that allow to circumvent the above men-tioned difficulties and obtain explicit bounds on precisionbased on QFI for arbitrary N , and in particular graspthe asymptotic precision scaling (Demkowicz-Dobrzański et al. , 2012; Escher et al. , 2011; Knysh et al. , 2014).These methods include: the minimization over channelpurifications method (Escher et al. , 2011) which is ap-plicable in general but requires some educated guess toobtain a useful bound, as well as classical and quantumsimulation methods (Demkowicz-Dobrzański et al. , 2012)which are applicable when the noise acts in an uncorre-lated manner on the probes, but have an advantage ofbeing explicit and convey some additional physical intu-ition on the bounds derived. Description of the newlypublished method (Knysh et al. , 2014) which is basedon continuous approximation of the probe states and thecalculus of variations is beyond the scope of this review.We will present the methods by applying them directlyto interferometry with each of the decoherence models in-troduced above. We invert the order of presentation ofthe bounds for the decoherence models compared withthe order in Sec. VI.A, as this will allow us to discussthe methods in the order of increasing complexity. Thesimplest of the methods, the classical simulation, will beapplied to the imperfect visibility model, while the quan-tum simulation will be discussed in the context of loss.Finally, minimization over channel purification methodwill be described in the context of the phase diffusionmodel, to which classical and quantum simulation meth-ods are not applicable due to noise correlations. Weshould note that methods of (Escher et al. , 2011; Knysh et al. , 2014) can also be successfully applied to uncorre-lated noise models. Still, classical and quantum simula-tion approaches are more intuitive and that is why wepresent derivations based on them even though the othertechniques yield equivalent bounds.In order to appreciate the significance of the derivedbounds, we will always compare them with the preci- sion achievable with a state of N uncorrelated photonsas given in Eq. (97). The ratio between this quantitiesbounds the amount of quantum-precision enhancementthat can be achieved with the help of quantum correla-tions present in the input state of N photons.
1. Imperfect visibility
The fundamental QFI bound on precision in case ofimperfect visibility or equivalently the local dephasingmodel has been derived in (Demkowicz-Dobrzański et al. ,2012; Escher et al. , 2011; Knysh et al. , 2014) and reads: ∆ ϕ ≥ s − η η √ N , (136)where η is the dephasing parameter, see Sec. VI.A.3.For the optimal uncorrelated input state, | ψ in i = [( | a i + | b i ) / √ ⊗ N we get ∆ ϕ = 1 / p η N , and hence the quan-tum precision enhancement which is the ratio of thebound on precision achievable for the optimal strat-egy and the precision for the uncorrelated strategy isbounded by a constant factor of p − η . a. Classical simulation method The derivation of the for-mula (136) presented below makes use of the classicalsimulation method (Demkowicz-Dobrzański et al. , 2012),which requires viewing the quantum channel represent-ing the action of the interferometer from a geometricalperspective. The set of all physical quantum channels,
Λ : L ( H in ) → L ( H out ) , that map between density matricesdefined on the input/ouput Hilbert spaces ( H in / out ) con-stitutes a convex set (Bengtsson and Zyczkowski, 2006).This is to say that given any two quantum channels Λ , Λ , their convex combination Λ = p Λ + (1 − p )Λ , ≤ p ≤ is also a legitimate quantum channel. Physi-cally Λ corresponds to a quantum evolution that is equiv-alent to a random application of Λ , Λ transformationswith probabilities p , − p respectively.As derived in Sec. VI.A.3, within the imperfect visibil-ity (local dephasing) decoherence model: ρ ϕ = Λ ⊗ Nϕ ( ρ in ) ,and hence the relevant quantum channel, has a simpletensor structure. Consider a single-photon channel Λ ϕ ,which ϕ -dependence we may depict as a trajectory withinthe set of all single-photon quantum maps, see Fig. 10.The question of sensing the parameter ϕ can now betranslated to the question of determining where on thetrajectory a given quantum channel Λ ϕ lies.Consider a local classical simulation (CS) of a quantumchannel trajectory Λ ϕ in the vicinity of a given point ϕ , ϕ = ϕ + δϕ (Demkowicz-Dobrzański et al. , 2012;Matsumoto, 2010), Λ ϕ [ ̺ ] = X x p ϕ ( x ) Π x [ ̺ ] + O (cid:0) δϕ (cid:1) , (137)0 FIG. 10 The space of all quantum channels, Λ , whichmap between density matrices specified on two given Hilbertspaces, Λ : L ( H in ) → L ( H out ) , represented as a convex set.The estimated parameter ϕ specifies a trajectory, Λ ϕ ( blackcurve ), in such a space. From the point of view of the QFI,any two channel trajectories, e.g. Λ ϕ and ˜Λ ϕ ( gray curve ), areequivalent at a given ϕ as long as they and their first deriva-tives with respect to ϕ coincide there. Moreover, they can beoptimally classically simulated at ϕ by mixing two channelslying on the intersection of the tangent to the trajectory andthe boundary of the set: Π ± . which represents the variation of the channel up tothe first order in δϕ as a classical mixture of some ϕ -independent channels { Π x } x where the ϕ dependence ispresent only in the mixing probabilities p ϕ ( x ) . Undersuch a construction the random variable X distributedaccording to p ϕ ( x ) specifies probabilistic choice of chan-nels Π x that reproduces the local action of Λ ϕ in thevicinity of ϕ . Crucially, the QFI is a local quantity—see discussions in Sec. IV.B.1—and at a given point ϕ is a function only of the quantum state considered andits first derivative with respect to the estimated param-eter. Consequently, when considering the parameter be-ing encoded in a quantum channel, all the channel tra-jectories at a given point ϕ are equivalent from thepoint of view of QFI if they lead to density matricesthat are identical up to the first order in δϕ . In otherwords we can replace Λ ϕ with any ˜Λ ϕ and obtain thesame QFI at a given point ϕ provided Λ ϕ = ˜Λ ϕ and d Λ ϕ d ϕ = d ˜Λ ϕ d ϕ (cid:12)(cid:12)(cid:12) ϕ = ϕ , see Fig. 10. This means that whenconstructing a local CS of the quantum channel Λ ϕ at ϕ , we need only to satisfy P x p ϕ ( x )Π x = Λ ϕ , as wellas P x d p ϕ ( x ) d ϕ | ϕ = ϕ Π x = d Λ ϕ d ϕ | ϕ = ϕ .Crucially, as the maps Λ ϕ in Eq. (135) act indepen-dently on each photon, we can simulate the overall Λ ⊗ Nϕ with N independent random variables, X N , associatedwith each channel. The estimation procedure can nowbe described as ϕ → X N → Λ ⊗ Nϕ → Λ ⊗ Nϕ [ | ψ in i ] → ˜ ϕ, . (138)where N classical random variables are employed to gen-erate the desired quantum map Λ ⊗ Nϕ . It is clear that astrategy in which we could infer the parameter directly from X N , i.e. ϕ → X N → ˜ ϕ , can perform only better thanthe scheme where the information about ϕ is firstly en-coded into the quantum channel which acts on the probestate and afterwards decoded from the measurement re-sults performed on the output state. This way, we mayalways construct a classically scaling lower bound on theprecision, or equivalently an upper bound on the QFI of ρ ϕ (135): F Q [ ρ ϕ ] ≤ F cl (cid:2) p Nϕ (cid:3) = N F cl [ p ϕ ] , (139)which is determined by the classical FI (54) evaluatedfor the probability distribution p ϕ ( X ) . Importantly,Demkowicz-Dobrzański et al. (2012) have shown that forthe estimation problems in which the parameter is unitar-ily encoded, it is always optimal to choose a CS depictedin Fig. 10, which employs for each ϕ only two channels Π ± lying at the points of intersection of the tangent to thetrajectory with the boundary of the quantum maps set.Such an optimal CS leads to the tightest upper boundspecified in Eq. (139): F Q [ ρ ϕ ] ≤ N/ ( ε + ε − ) , where ε ± are the “distances” to the boundary marked in Fig. 10, Π ± = Λ ϕ ± ε ± d Λ ϕ d ϕ | ϕ = ϕ .Looking for ε ± parameters amounts to a search ofthe distances one can go along the tangent line to thetrajectory of Λ ϕ so that the corresponding map is stilla physical quantum channel, i.e. a completely posi-tive trace preserving map. This is easiest to do mak-ing use of the Choi-Jamiolkowski isomorphism (Choi,1975; Jamiołkowski, 1972) which states that with eachquantum channel, Λ : L ( H in ) → L ( H out ) , we can asso-ciate a positive operator Ω Λ ∈ L ( H out ⊗ H in ) , so that Ω Λ = (Λ ⊗ I )( | I ih I | ) , where | I i = P dim ( H in ) i =1 | i i| i i is amaximally entangled state on H in ⊗ H in , while I is theidentity map on L ( H in ) . Since the complete positivity of Λ is equivalent to positivity of the Ω Λ operator, one needsto analyze Ω Λ ϕ ± ε ± d Ω Λ ϕ d ϕ | ϕ = ϕ and find maximum ε ± so that the above operator is still positive-semidefinite.Taking the explicit form of the Λ ϕ for the caseof optical interferometry with imperfect visibility, seeSec. VI.A.3, one can show that ε ± = p − η /η (Demkowicz-Dobrzański et al. , 2012), which yields theultimate quantum limit on precision given by Eq. (136).
2. Photonic losses
The expression for the QFI of the output state (127)in the asymptotic limit of large N has been first derivedby (Knysh et al. , 2011). Yet, the general frameworksproposed by (Demkowicz-Dobrzański et al. , 2012; Escher et al. , 2011) for generic decoherence allowed to recon-struct this bound on precision with the following result: ∆ ϕ ≥ (cid:18)r − η a η a + r − η b η b (cid:19) √ N , . (140)1 ̺ ̺ ϕ ̺ σ ϕ Φ ̺ ϕ FIG. 11 The Quantum Simulation (QS) of a channel. The ac-tion of the channel Λ ϕ is simulated up to the first order in thevicinity of a given point ϕ using a ϕ -independent Φ channeland an auxiliary state σ ϕ that contains the full informationabout the estimated parameter ϕ . where η a , η b are transmission in the two arms of the inter-ferometer respectively, see VI.A.2. This bound simplifiesto ∆ ϕ ≥ r − ηηN (141)in the case of equal losses, and since the precision achiev-able with uncorrelated states is given by / √ ηN , themaximal quantum-enhancement factor is √ − η . In thefollowing, we derive the above bounds using the quantumsimulation approach of (Demkowicz-Dobrzański et al. ,2012; Kołodyński and Demkowicz-Dobrzański, 2013). a. Quantum Simulation method Unfortunately, in thecase of loss the simple CS method yields a trivial bound ∆ ϕ ≥ , since the tangent distances to the boundary ofthe set of quantum channels are ε ± = 0 in this case. It ispossible, however, to derive a useful bound via the Quan-tum Simulation (QS) method which is a natural gener-alization of the CS method. The QS method has beendescribed in detail and developed for general metrologi-cal schemes with uncorrelated noise by Kołodyński andDemkowicz-Dobrzański (2013) stemming from the worksof Demkowicz-Dobrzański et al. (2012) and Matsumoto(2010).As shown in Fig. 11, local QS amounts to re-expressingthe action of Λ ϕ for ϕ = ϕ + δϕ by a larger ϕ - independent map Φ that also acts on the auxiliary ϕ - dependent input σ ϕ , up to the first order in δϕ : Λ ϕ [ ̺ ] = Tr E Φ [ σ ϕ ⊗ ̺ ] + O ( δϕ ) . (142)Note that for σ ϕ = P x p ϕ ( x ) | x ih x | , and Φ = | x ih x | ⊗ Π x , QS becomes equivalent to the CS of Eq. (137), sothat CS is indeed a specific instance of the more generalQS. An analogous reasoning as in the case of CS leads tothe conclusion that we may upper-bound the QFI of theoverall output state, here (127) for the case of losses, as F Q (cid:2) Λ ⊗ Nϕ [ | ψ i in h ψ | ] (cid:3) = F Q (cid:2) Tr E (cid:8) Φ ⊗ N (cid:2) σ ⊗ Nϕ ⊗ | ψ i in h ψ | (cid:3)(cid:9)(cid:3) ≤ F Q (cid:2) σ ⊗ Nϕ (cid:3) = N F Q [ σ ϕ ] , since Tr E Φ ⊗ N [ · ] is just a parameter independent map,under which the overall QFI may only decrease—see Eq. (82). Last equality follows from the additivity prop-erty of the QFI, which, similarly to Eq. (139), constrains F Q (cid:2) ρ Nϕ (cid:3) to scale at most linearly for large N . Similarlyto the case of CS, in order to get the tightest bound oneshould find QS that yields the smallest F Q [ σ ϕ ] , which inprinciple is a non trivial task.Fortunately, Kołodyński and Demkowicz-Dobrzański(2013) have demonstrated that the search for the opti-mal channel QS corresponds to the optimization over theKraus representation of a given channel. Without lossof generality we may assume that σ ϕ = | ϕ ih ϕ | is a pure ϕ -dependent state while Φ[ · ] = U · U † is unitary. For agiven QS we may write the corresponding Kraus repre-sentation of the channel by choosing a particular basis | i i E in the E space: K i ( ϕ ) = E h i |U| ϕ i E . In order forthe QS to be valid, these Kraus operators should corre-spond to a legitimate Kraus representation of the channel Λ ϕ [ · ] = P i K i ( ϕ ) · K i ( ϕ ) . Two Kraus representation of agiven quantum channel are equivalent if and only if theyare related by a unitary matrix u : ˜ K i ( ϕ ) = X j u ij ( ϕ ) K j ( ϕ ) , (143)which may in principle be also ϕ dependent. Since werequire QS to be only locally valid in the vicinity of ϕ ,the above equation as well as its first derivative needsto be fulfilled only at ϕ . Because of that, the searchfor the optimal Kraus representation ˜ K i (or equivalentlythe optimal QS) may be restricted to the class of trans-formations where u ( ϕ ) = e i( ϕ − ϕ ) h with h being anyHermitian matrix that shifts the relevant derivative of K i ( ϕ ) at ϕ , so that ˜ K i ( ϕ ) = K i ( ϕ ) and ˙˜ K i ( ϕ ) =˙ K i ( ϕ ) + i P j h ij K j ( ϕ ) . As shown by (Kołodyński andDemkowicz-Dobrzański, 2013), the problem of finding theoptimal QS i.e. the minimal F Q [ | ϕ ih ϕ | ] which we termas the F QS , can be formally rewritten as F QS = min h s s.t. X i ˙˜ K i ( ϕ ) † ˙˜ K i ( ϕ ) = s , X i ˙˜ K i ( ϕ ) † ˜ K i ( ϕ ) = , (144)where the parameter s has the interpretation of F Q [ | ϕ ih ϕ | ] for the particular QS at ϕ and the constraintsimposed in Eq. (144) are necessary and sufficient for theQS required transformation U and the state | ϕ i to exist.The above optimization problem may not always beeasy to solve. Still, its relaxed version: min h k X i ˙˜ K i ( ϕ ) † ˙˜ K i ( ϕ ) k , X i ˙˜ K i ( ϕ ) † ˜ K i ( ϕ ) = , (145)where k · k is the operator norm, can always be cast in theform of an explicit semi-definite program, which can beeasily solved numerically (Demkowicz-Dobrzański et al. ,2012). Numerical solution of the semi-definite program2provides a form of the optimal h which may then be takenas an ansatz for further analytical optimization.Plugging in the Kraus operators K i ( ϕ ) = K i U ϕ repre-senting the lossy interferometer, see Eq. (130), and fol-lowing the above described procedure one obtains F QS = 4 (cid:16)q − η a η a + q − η b η b (cid:17) (146)for the optimal h given by h opt = − diag (cid:26) χ, η a − η a (cid:18) η a − χ (cid:19) , − η b − η b (cid:18) η a + χ (cid:19)(cid:27) , (147)where χ = F QS η b − η a η a η b . This indeed reproduces the boundsgiven in (140).In order to provide the reader with a simple intuitionconcerning the QS method, we shall present an elemen-tary construction of the QS for lossy interferometer inthe special case of η a = η b = 1 / . In this case the bound(140) yields ∆ ϕ ≥ / √ N which implies that using opti-mal entangled probe state at the input under lossescannot beat the precision which can be obtained by un-correlated probes in ideal scenario of no losses.Consider the action of the single photon lossy channel Λ ϕ on the pure input state | ψ i = α | a i + β | b i : Λ ϕ ( | ψ ih ψ | ) = 12 | ψ ϕ ih ψ ϕ | + 12 | vac ih vac | (148)with | ψ ϕ i = α e i ϕ | a i + β | b i , which represents / proba-bility of photon sensing the phase undisturbed and the / probability of the photon being lost. Let the auxil-iary state for QS be | ϕ i = (e i ϕ | i + | i ) / √ . The joinedinput + auxiliary state reads: | φ i| ψ i = 1 √ (cid:0) α e i ϕ | i| a i + β | i| b i (cid:1) ++ 1 √ (cid:0) α e i ϕ | i| b i + β | i| a i (cid:1) . (149)The map Φ realizing the QS consists now of two steps.First the controlled NOT operation is performed with theauxiliary system being the target qubit, this transformthe above state to: √ | i ( α e i ϕ | a i + β | b i )+ √ | i ( α e i ϕ | b i + β | a i ) . The second step is the measurement of the aux-iliary system. If the result | i is measured (probability / ), the system is left in the correct state | ψ ϕ i and themap leaves it unchanged, if the | i is measured the stateof the photon is not the desired one, in which case themap returns the | vac i state. This map is therefore aproper QS of the desired lossy interferometer transfor-mation for η a = η b = 1 / . Since the auxiliary state em-ployed in this construction was | ϕ i = (e i ϕ | i + | i ) / √ forwhich F Q ( | ϕ ih ϕ | ⊗ N ) = N this leads to the anticipatedresult ∆ ϕ ≥ / √ N .
3. Phase diffusion
Since the phase diffusion model, see Sec. VI.A.1, isan example of a correlated noise model, it cannot be ap-proached with the CS and QS methods. The study of thebehavior of the QFI within the phase-diffusion model wasfor the first time carried out by (Genoni et al. , 2011) con-sidering indefinite-photon-number Gaussian input statesand studied numerically the achievable precision and thestructure of optimal input states. Yet, the fundamen-tal analytical bounds on precision have not been veri-fied until Escher et al. (2012), where the phase noise hasbeen approached using the minimization over purifica-tions method of Escher et al. (2011) and most recentlyusing the calculus of variations approach of Knysh et al. (2014). a. Minimization over purification method
The minimiza-tion over purification method of Escher et al. (2011)is based on the observation, already mentioned inSec. IV.B.1, that QFI for a given mixed quantum state,here ρ ϕ (127), is not only upper bounded by the QFI ofany of its purifications, but there always exists an opti-mal purification, (cid:12)(cid:12) Ψ opt ϕ (cid:11) , for which F Q [ ρ ϕ ] = F Q (cid:2)(cid:12)(cid:12) Ψ opt ϕ (cid:11)(cid:3) ,where ρ ϕ = Tr E (cid:8)(cid:12)(cid:12) Ψ opt ϕ (cid:11)(cid:10) Ψ opt ϕ (cid:12)(cid:12)(cid:9) . As such statement doesnot rely on the form of the transformation | ψ i in → ρ ϕ but rather on the properties of the output state itself,the framework of Escher et al. (2011) in principle doesnot put any constraints on the noise-model considered.Note that, even if the optimal purification itself is diffi-cult to find, any suboptimal purification yields a legiti-mate upper bound on the QFI and hence may provide anon-trivial precision bound.In order to get a physical intuition regarding the pu-rification method, consider a physical model of the phasediffusion where light is being reflected from a mirrorwhich position fluctuations are randomly changing theeffective optical length. Formally, the model amounts tocoupling the phase delay generator ˆ J z to the mirror posi-tion quadrature ˆ x E = √ (cid:16) ˆ a E + ˆ a † E (cid:17) (Escher et al. , 2012).Assuming the mirror, serving as the environment E, toreside in the ground state of a quantum oscillator | i E before interaction with the light beam, the pure outputstate reads: | Ψ ϕ i = e − i ϕ ˆ J z e i √
2Γ ˆ J z ˆ x E | ψ i in | i E . (150)Thanks to the fact that |h x | i| = e − x / √ π , the reducedstate ρ ϕ = ∞ ˆ −∞ d x E h x | Ψ ϕ ih Ψ ϕ | x i E (151)indeed coincides with the correct output phase diffusedstate (125). Therefore, this is a legitimate purification3of the interferometer output state in presence of phasediffusion.Consider now another purification (cid:12)(cid:12)(cid:12) ˜Ψ ϕ E = e i δϕ ˆ H E | Ψ ϕ i generated by a local ( ϕ = ϕ + δϕ ) rotation of the mirrormodes, i.e. a unitary operation on the system E. We lookfor a transformation of the above form which hopefullyerases as much information on the estimated phase aspossible, so that QFI for the purified state (cid:12)(cid:12)(cid:12) ˜Ψ ϕ E will beminimized leading to the best bound on the QFI of ρ ϕ .Choosing ˆ H E = λ ˆ p E we obtain the following upper boundon the QFI F Q [ ρ ϕ ] ≤ min λ (cid:8) F Q (cid:2) e i ϕλ ˆ p E | Ψ( ϕ ) i (cid:3)(cid:9) == min λ (cid:26) λ + 4 (cid:16) − √ λ (cid:17) ∆ J z (cid:27) = 4 ∆ J z J z , and thus a lower limit on the precision ∆ ϕ ≥ r Γ + 14 ∆ J z = r Γ + 1 N , (152)where we plugged in ∆ J z = N / corresponding tothe NOON state which maximized the variance for N -photon states. See also an alternative derivation of theabove result that has been proposed recently in (Ma-cieszczak, 2014). Crucially, the above result proves thatthe phase diffusion constrains the error to approach aconstant value √ Γ as N → ∞ , which does not vanishin the asymptotic limit, what contrasts the / √ N be-havior characteristic for uncorrelated noise models. Notealso that due to the correlated character of the noise,the bound (152) predicts that it may be more benefi-cial to perform the estimation procedure on a group of k particles and then repeat the procedure independently ν times obtaining / √ ν reduction in estimation error,rather than employing N = kν in a single experimentalshot (Knysh et al. , 2014).Only very recently, the exact ultimate quantum limitfor the N -photon input states has been derived in (Knysh et al. , 2014) ∆ ϕ ≥ r Γ + π N , (153)showing that the previous bound was not tight, with thesecond term following the HL-like asymptotic scaling ofthe noiseless decoherence-free Bayesian scenario statedin Eq. (107). In fact, as proven by (Knysh et al. , 2014),the optimal states of the noiseless Bayesian scenario, i.e.the sine states (105), attain the above correct quantumlimit. In Sec. VI.C.2, we show that within the Bayesianapproach, with the phase-diffusion effects incorporated,the sine states are always the optimal inputs. C. Bayesian approach
Minimizing the average Bayesian cost, as given byEq. (93), over input probe states | ψ i in is in general moredemanding than minimization of the QFI due to the factthat it is not sufficient to work within the local regimeand analyze only the action of a channel and its firstderivative at a given estimation point as in the QFI ap-proach. For this reason we do not apply the Bayesianapproach to the imperfect visibility model as obtainingthe bounds requires a significant numerical and analyt-ical effort (Jarzyna and Demkowicz-Dobrzanski, 2014),and constrain ourselves to loss and phase diffusion mod-els.
1. Photonic losses
The optimal Bayesian performance of N -photonstates has been studied by Kołodyński and Demkowicz-Dobrzański (2010). Assuming the natural cost function(67) and the flat prior phase distribution, the averagecost (93) reads: h C i = Tr (cid:8) h ρ ϕ i C Ξ (cid:9) (154)where h ρ ϕ i C = 4 ´ dϕ π ρ ϕ sin (cid:0) ϕ (cid:1) and ρ ϕ is given by (127).The optimal measurement seed operator Ξ can be foundanalogously as in the decoherence-free case. The blockdiagonal form of ρ ϕ , implies that without losing op-timality one can assume Ξ = L NN ′ =0 | e N ′ ih e N ′ | with | e N ′ i = P N ′ n =0 | n, N ′ − n i . Physically, the block-diagonalstructure of Ξ indicates that the optimal covariant mea-surement requires a non-demolition photon number mea-surement to be performed before carrying out any phasemeasurements, so that the orthogonal subspaces, labeledby the number of surviving photons N ′ , may be firstlydistinguished, and subsequently the measurement whichis optimal in the lossless case is performed (Kołodyńskiand Demkowicz-Dobrzański, 2010). Plugging in the ex-plicit form of Ξ together with the explicit form of theoutput state ρ ϕ , we arrive at h C i = 2 − c T Ac , A n,n − = A n − ,n = n,N − n X l a ,l b =0 q b ( l a ,l b ) n b ( l a ,l b ) n − , (155)where A is a symmetric ( N +1) × ( N +1) matrix that isnon-zero only on its first off-diagonals, b ( l a ,l b ) n are the bi-nomial coefficients previously defined in Eq. (128), while c is a state vector containing coefficients c n of the N -photon input state (120).The minimal average cost (154) for the lossy interfer-ometer then equals h C i min = 2 − λ max , where λ max isthe maximal eigenvalue of the matrix A and the cor-responding eigenvector c max provides the optimal input4state coefficients. h C i min quantifies the maximal achiev-able precision and in the N → ∞ limit may be interpretedas the average MSE (60) due to the convergence of thecost function (67) to the squared distance as ˜ ϕ → ϕ .The procedure described above allows only to obtainnumerical values of the achievable precision, and ceases tobe feasible for N → ∞ . The main result of (Kołodyńskiand Demkowicz-Dobrzański, 2010) was to construct avalid analytical lower bound on the minimal average cost(154): h C i min ≥ (cid:20) − A max cos (cid:18) πN + 2 (cid:19)(cid:21) , (156)where A max = max ≤ n ≤ N { A n,n − } is the largest elementof the matrix A , contained within its off-diagonal entries(155). The bound yields exactly the same formula as theQFI bound (140), proving that in this case the Bayesianand QFI approaches are equivalent: ∆ ϕ ≈ p h C i ≥ q h C i min ≥ (cid:18)r − η a η a + r − η b η b (cid:19) √ N , (157)where ≈ represents the fact that Bayesian cost approxi-mated the variance only in the limit of large N . The factthat both approaches lead to the same ultimate boundson precision suggests that the optimal input states maybe approximated for N → ∞ up to an arbitrary good pre-cision with states manifesting only local finite-number ofparticle correlations and may in particular be efficientlysimulated with the concept of matrix-product states(Jarzyna and Demkowicz-Dobrzański, 2013; Jarzyna andDemkowicz-Dobrzanski, 2014).
2. Phase diffusion
Similarly to the case of losses discussed in the previ-ous section, we study the estimation precision achievedwithin the Bayesian approach but in the presence of phasediffusion . The analysis follows exactly in the same way,so that likewise assuming no prior knowledge and thenatural cost function introduced in Eq. (93) the formulafor the average cost reads h C i = Tr (cid:8) h ρ ϕ i C Ξ (cid:9) = 2 − c T Ac , (158)where this time one may think of the effective state, asof the input state ρ in which is firstly averaged over theGaussian distribution dictated by the evolution (124) andthen over the cost function in accordance with Eq. (93).The optimal seed element of the covariant POVM is iden-tical as in the decoherence-free case Ξ = | e N ih e N | andthe matrix A possesses again only non-zero entries on itsfirst off-diagonals, but this time all of them are equal toe − Γ2 . As a result, the minimal average cost (158) may beevaluated analytically following exactly the calculation of (Berry and Wiseman, 2000) for the noiseless scenario,which leads then to λ max = 2 e − Γ2 cos (cid:16) πN +2 (cid:17) and hence h C i min = 2 (cid:20) − e − Γ2 cos (cid:18) πN + 2 (cid:19)(cid:21) N →∞ ≈ (cid:16) − e − Γ2 (cid:17) + e − Γ2 π N . (159)The optimal input states are the same as in thedecoherence-free case, i.e. they are the N -photon sinestates of Eq. (105). Note that in contrast to the pho-tonic loss which is an example of an uncorrelated noise,the minimal average cost (159) does not asymptoticallycoincide with the QFI-based precision limit (153) unless Γ ≪ . D. Practical schemes saturating the bounds
Deriving the fundamental bounds on quantum en-hanced precision in presence of decoherence is interest-ing in itself from a theoretical a point of view. Still,a practical question remains whether the bounds de-rived are saturable in practice. Note that NOON statesand the sine states that are optimal in case of QFI andBayesian approaches in the decoherence-free case arenotoriously hard to prepare apart from regime of verysmall N . For large photon numbers, the only practicallyaccessible states of light are squeezed Gaussian statesand one of the most popular strategies in performingquantum-enhanced interferometry amounts to mixing acoherent beam with a squeezed vacuum state on the in-put beam splitter of the Mach-Zehnder interferometer,see Sec. III.D. We demonstrate below that in presenceof uncorrelated decoherence, such as loss or imperfectvisibility, this strategy is indeed optimal in the asymp-totic regime of large N and allows to saturate the fun-damental bounds derived above. We will not discuss thephase-diffusion noise, since the estimation uncertainty isfinite in the asymptotic limit, and the issue of saturat-ing the asymptotic bound becomes trivial as practicallyall states lead to the same asymptotic precision value,while saturating the bound for finite N requires the useof experimentally inaccessible sine states.
1. Bounds for indefinite photon number states
Derivation of the bounds presented in this section bothin the QFI and Bayesian approaches assumed definite -photon number states at the input. We have already dis-cussed the issue of translating the bounds from a definitephoton number input state case to a general indefinite-photon number state case in Sec. V.C in the case ofdecoherence-free metrology, where we have observed that5due to quadratic dependence of QFI on number of pho-tons used, maximization of QFI over states with fixed averaged photon number h N i is ill defined and arbitraryhigh QFI are in principle achievable. Controversies re-lated to this observation, discussed in Sec. V.C.2, arefortunately not present in the noisy metrology scenario.For the decoherence models, analyzed in this paper,the QFI scales at most linearly with N . Following thereasoning presented in Sec. V.C, consider a mixture ofdifferent photon number states P N p N ρ N . Since in thepresence of decoherence F Q ( ρ N ) ≤ cN , where c is a con-stant coefficient that depends on the type and strengthof the noise considered, thanks to the convexity of theQFI we can write: F Q X N p N ρ N ! ≤ X N p N F Q ( ρ N ) ≤ X p N cN = c h N i . (160)Hence the bounds on precision derived in Sec. VI.B.2(losses) and Sec. VI.B.1 (imperfect visibility) are validalso under replacement of N by h N i . Still, one may comeacross claims of precisions going beyond the above men-tioned bounds typically by a factor of two (Aspachs et al. ,2009; Joo et al. , 2011). This is only possible, however,if classical reference beam required to perform e.g. thehomodyne detection is not treated as a resource. As dis-cussed in detail in Sec. V.C.1, we take the position thatsuch reference beams should be treated in the same wayas the light traveling through the interferometer and assuch also counted as a resource.
2. Coherent + squeezed vacuum strategy
In section Sec. III.A, we have derived an error-propagation formula for the phase-estimation uncertainty(33) for the standard Mach-Zehnder interferometry in ab-sence of decoherence. For this purpose we have adoptedthe Heisenberg picture and expressed the precision interms of expectation values, variances and covariances ofthe respective angular momentum observables calculatedfor the input state. Here, we follow the same procedurebut take additionally into account the effect of imperfectvisibility (local dephasing) and loss. For simplicity, inthe case of loss we restrict ourselves to equal losses inboth arms. The Heiseberg picture transformation of anobservable ˆ O corresponding to a general map Λ ϕ (123)reads X i U † ϕ K † i ˆ OK i U ϕ = Λ ∗ ϕ ( ˆ O ) , (161)where Λ ∗ is called the conjugated map.For a more direct comparison with the decoherence-free formulas of Sec. III.A, we explicitly include the actionof the Mach-Zehnder input and output balanced beam splitters in the description of the state transformation—in terms of Fig. 8 this corresponds to moving | ψ i in tothe left and ρ ϕ to the right of the figure. In case ofloss the decoherence map has the same form as given inSec. VI.A.2, but with U ϕ = e − i ϕ σ y , while in the caseof imperfect visibility the Kraus operators (134) will bemodified to K = q η K = q − η σ y , so that the lo-cal dephasing is defined with respect to the y rather thanthe z axis. For the two decoherence models, the resultingHeisenberg picture transformation of the J z observableyields (Ma et al. , 2011): h ˆ J z i η = cos ϕ h ˆ J z i in − sin ϕ h ˆ J x i in , (162) ∆ J z η = f ( η ) h N i ϕ ∆ J z | in + sin ϕ ∆ J x | in + − ϕ cos ϕ cov ( J x , J z ) | in . where f ( η ) = (1 − η ) /η for the loss model and f ( η ) =(1 − η ) /η in the case of local dephasing model. Theabove expressions have a clear intuitive interpretation.The signal h ˆ J z i is rescaled by a factor η compared withthe decoherence-free case, while the variance apart fromthe analogous rescaling is enlarged by an additional noisecontribution f ( η ) h N i / due to lost or dephased photons.In order to calculate the precision achievable withcoherent+squeezed-vacuum strategy, we may use the al-ready obtained quantities presented in Eq. (37). Af-ter substituting the input variances and averages intoEq. (162) and optimally setting α = Re ( α ) as before, wearrive at a modified version of the formula (38) for thephase estimation precision: ∆ ϕ | α i| r i == r cot ϕ ( | α | + sinh r ) + | α | e − r +sinh r + f ( η ) | α | r sin2 ϕ | | α | − sinh r | . (163)The optimal operation points are again ϕ = π/ , π/ .Considering the asymptotic limit h N i = | α | + sinh r →∞ and assuming the coherent beam to carry the dom-inant part of the energy | α | ≫ sinh r , the formula forprecision at the optimal operation point reads: ∆ ϕ | α i| r i ≈ p h N i e − r + f ( η ) h N ih N i = p e − r + f ( η ) p h N i . (164)Clearly, even for relatively small squeezing strength r the e − r term becomes negligible, and hence we can effec-tively approach arbitrary close precision given by: ∆ ϕ | α i| r i ≈ p f ( η ) p h N i , (165)which recalling the definition of f ( η ) for the two de-coherence models considered coincides exactly with the6 ∆ ϕ ≥ √ N ∆ ϕ ≤ N FIG. 12 The phase estimation precision of an interferome-ter with equal losses in both arms ( η = 0 . ). The perfor-mance of the optimal N -photon input states (120) is shown( solid black ) that indeed saturate the asymptotic quantumlimit (140) ( dotted ): p (1 − η ) / ( ηN ) . The NOON states( solid grey ) achieve nearly optimal precision only for low N ( ≤ ) and rapidly diverge becoming out-performed by classi-cal strategies. For comparison, the precision attained for anindefinite photon number scheme is presented, i.e. a coher-ent state and squeezed vacuum optimally mixed on a beam-splitter (Caves, 1981) ( dashed ), which in the presence of lossalso saturates the asymptotic quantum limit (140). fundamental bounds (136), (141) derived before. Thisproves that the fundamental bounds can be asymptoti-cally saturated with a practical interferometric scheme.One should note that this contrasts the noiseless case andthe suboptimal performance of simple estimation schemebased on the photon-number difference measurements,see Eq. (39).To summarize the results obtained in this section, inFig. 12, we present a plot of the maximal achievable pre-cision for the lossy interferometer in the equal-losses sce-nario with η = 0 . , i.e. ∆ ϕ = 1 / p ¯ F Q [ ρ ϕ ] as a functionof N compared with the NOON state–based strategy aswell as the asymptotic bound (140). On the one hand,the NOON states remain optimal for relatively small N ( ≤ ), for which the effects of losses may be disre-garded. This fact supports the choice of NOON -likestates in the quantum-enhanced experiments with smallnumber of particles (Krischek et al. , 2011; Mitchell et al. ,2004; Nagata et al. , 2007; Okamoto et al. , 2008; Resch et al. , 2007; Xiang et al. , 2010). However, one shouldnote that in the presence of even infinitesimal losses, theprecision achieved by the NOON states quickly divergeswith N , because their corresponding output state QFI, F NOONQ = η N N , decays exponentially for any η < .Most importantly, it should be stressed that the co-herent+squeezed vacuum strategy discussed above hasbeen implemented in recent gravitational-wave interfer-ometry experiments (LIGO Collaboration, 2011, 2013).The main factor limiting the quantum enhancement ofprecision in this experiments is loss, which taking into account detection efficiency, optical instruments imper-fections and imperfect coupling was estimated at the levelof (LIGO Collaboration, 2011). In (Demkowicz-Dobrzański et al. , 2013) it has been demonstrated thatthe sensing precision achieved in (LIGO Collaboration,2011) using the dB squeezed vacuum (correspondingto the squeezing factor e − r ≈ . ), was strikingly closeto the fundamental bound, and only further reduc-tion in estimation uncertainty would be possible if moreadvanced input states of light were used. VII. CONCLUSIONS
In this review we have showed how the tools of quan-tum estimation theory can be applied in order to derivefundamental bounds on achievable precision in quantum-enhanced optical interferometric experiments. The mainmessage to be conveyed is the fact that while the power ofquantum enhancement is seriously reduced by the pres-ence of decoherence, and in general the Heisenberg scal-ing cannot be reached, non-classical states of light of-fer a noticeable improvement in interferometric precisionand simple experimental schemes may approach arbi-trary close the fundamental quantum bounds. It is alsoworth noting that in the presence of uncorrelated deco-herence the Bayesian approaches coincide asymptoticallywith the QFI approaches easing the tension between thistwo often competing ways of statistical analysis.We would also like to mention an inspiring alterna-tive approach to the derivation of limits on precisionof phase estimation, where the results are derived mak-ing use of information theoretic concepts such as rate-distortion theory (Nair, 2012) or entropic uncertaintyrelations (Hall and Wiseman, 2012). Even though thebounds derived in this way are weaker than the boundspresented in this review and obtained via Bayesian orQFI approaches, they carry a conceptual appeal encour-aging to look for deeper connections between quantumestimation and communication theories.Let us also point out, that while we have focused ourdiscussion on optical interferometry using the paradig-matic Mach-Zehnder model, the same methods can beapplied to address the problems of fundamental precisionbounds in atomic interferometry (Cronin et al. , 2009),magnetometry (Budker and Romalis, 2007), frequencystabilization in atomic clocks (Diddams et al. , 2004) aswell as the limits on resolution of quantum enhancedlithographic protocols (Boto et al. , 2000). All these se-tups can be cast into a common mathematical frame-work, see Sec. III.F, but the resulting bounds will dependstrongly on the nature of dominant decoherence effectsand the relevant resource limitations such as: total ex-perimental time, light power, number of atoms etc., aswell as on the chosen figure of merit. In particular, it isnot excluded that in some atomic metrological scenarios7one may still obtain a better than / √ N of precision ifdecoherence is of a special form allowing for use of thedecoherence-free subspaces (Dorner, 2012; Jeske et al. ,2013) or when its impact may be significantly reduced byconsidering short evolution times, in which the SQL-likescaling bounds may in principle be circumvented by: ad-justing decoherence geometry (Arrad et al. , 2014; Chaves et al. , 2013; Dür et al. , 2014; Kessler et al. , 2013) or byconsidering Non-Markovian short-time behaviour (Chin et al. , 2012; Matsuzaki et al. , 2011).We should also note that application of the tools pre-sented in this review to a proper analysis of fundamen-tal limits to the operation of quantum enhanced atomicclocks (Andrè et al. , 2004; Leibfried et al. , 2004) is notthat direct as it requires taking into account precise fre-quency noise characteristic of the local oscillator, allow-ing to determine the optimal stationary operation regimeof the clock (Macieszczak et al. , 2013) ideally in termsof the Allan variance taken as a figure of merit (Fraas,2013). A deeper theoretical insight into this problem isstill required to yield computable fundamental bounds.The applicability of the tools presented has also beenrestricted to single phase parameter estimation. A moregeneral approach may be taken, were multiple-phases(Humphreys et al. , 2013) or the phase as well as thedecoherence strength itself are the quantities to be es-timated (Crowley et al. , 2014; Knysh and Durkin, 2013).This poses an additional theoretical challenge as thenthe multi-parameter quantum estimation theory needsto be applied, while most of the tools discussed in thisreview are applicable only to single-parameter estima-tion. Developing non-trivial multi-parameter fundamen-tal bounds for quantum metrology is therefore still anopen field for research.This research work supported by the FP7 IP projectSIQS co-financed by the Polish Ministry of Science andHigher Education, Polish NCBiR under the ERA-NETCHIST-ERA project QUASAR, and Foundation for Pol-ish Science TEAM project. REFERENCES
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