aa r X i v : . [ qu a n t - ph ] F e b Quantum metrology and coherence
Laura Ares ∗ and Alfredo Luis † Departamento de ´Optica, Facultad de Ciencias F´ısicas, Universidad Complutense, 28040 Madrid, Spain (Dated: February 9, 2021)We address the relation between quantum metrological resolution and coherence. We examinethis dependence in two manners: we develop a quantum Wiener-Kintchine theorem for a suitablemodel of quantum ruler, and we compute the Fisher information. The two methods have the virtueof including both the contributions of probe and measurement on an equal footing. We illustrate thisapproach with several examples of linear and nonlinear metrology. Finally, we optimize resolutionregarding coherence as a finite resource.
I. INTRODUCTION
Coherence is a basic and elusive concept in classicaloptics, as well as in modern quantum mechanics andquantum optics. Coherence is not a phenomenon by it-self but it underlies the most relevant phenomena of op-tics and quantum mechanics, including current quantumtechnologies for which decoherence is the most fearfulenemy [1]. In recent times it has been recognized thefundamental role of coherence as a quantum resource forthe most promising applications of the quantum theory[2].There have been many different theories formalisingthe concept of coherence and its practical consequences inquantum physics, from the earliest theory of Glauber, tothe most recent in the form of quantum resource theories[2, 3].Among the areas of application of coherence we caninclude quantum metrology, in which the effort to per-form the most careful observation of physical phenom-ena meets the limits imposed by the most bizarre conse-quences of quantum physics, involving basic issues suchas state reduction and uncertainty relations [4].In this work we elaborate a fresh relation betweenquantum resolution and coherence to be derived fromfirst principles. We propose a model of detection in whichthe measurement takes the form of a quantum ruler con-structed by the same group of transformations that en-codes the signal to be detected on the probe state. Thisleads to a resolution-coherence relation that fully mim-ics the classic Wiener-Kintchine theorem allowing a com-plete parallelism between quantum and classical coher-ence theories [5]. This approach is completed with themore usual formalism based on the quantum Cram´er–Rao bound in terms of Fisher information. We illustratethis approach by applying it to several relevant examples ∗ Electronic address: [email protected] † Electronic address: alluis@fis.ucm.es;URL: of linear and nonlinear metrology, looking as far as pos-sible for a physical understanding of the results obtained[6].A distinctive feature of the coherence–resolution rela-tion developed in this work is that it includes the quan-tum coherence properties of the detection process on anequal footing with the coherence of the probe state. Wethink this is relevant, specially regarding the optimiza-tion of resolution given the always limited finite amountof resources, where in the most common approaches onlyconsider the resources consumed by the probe.Our analysis confirms compelling evidence showingthat the true resource for precise detection is coherence.In this spirit we look for the optimum resolution given thefinite coherence resources conveyed by probe and mea-surement.
II. QUANTUM RULER
We base most of our analysis on a rather natural andparadigmatic model of measurement, where the signal todetected shifts the pointer against a ticked ruler in whichthe ticks are actually repeated shifts of a basic mark.The pointer is a quantum state with density ma-trix ρ , which is transformed by a signal-dependentunitary transformation D ( λ ) = exp( − iλG ) to ρ ( λ ) = D ( λ ) ρ D † ( λ ), where λ represents the signal to be de-tected, G is the infinitesimal generator of the transfor-mation, and we have D † ( λ ) = D ( − λ ).The ticked ruler is represented by the measurementperformed, to be called M , to be described by a positiveoperator-valued measure ∆( m ), where the outcomes m represent the ticks of the ruler. We consider that theticks are spaced by effect of the same transformation thatshifts the probe, this is∆( m ) = D ( m )∆ D † ( m ) , ∆ = ∆(0) , (2.1)for the same unitary transformation D ( m ) above, and ∆ is the origin for the ticks. For definiteness and withoutloss of generality we will assume that m is a continuousvariable extending from −∞ to ∞ .Among other interesting properties, this structureguarantees that we can derive general conclusions aboutthe performance of the detection process independentboth the of the signal value λ and of the outcome m ob-tained. This is clearly manifest by computing the statis-tics of the measurement p ( m | λ ) conditioned to the signalvalue λ , this is p ( m | λ ) = tr [ ρ ( λ )∆( m )] , (2.2)this is p ( m | λ ) = tr (cid:2) D ( λ ) ρ D † ( λ ) D ( m )∆ D † ( m ) (cid:3) , (2.3)leading to p ( m | λ ) = tr (cid:2) ρ D ( m − λ )∆ D † ( m − λ ) (cid:3) = p ( m − λ ) . (2.4)This conditional probability, along with the prior infor-mation we might have about λ , in the case there is any,is the building block of a complete signal-estimation pro-cess, for example within a Bayesian formulation [7, 11].A key result is that the conditional probability dependson λ and m just through the combination m − λ , this is p ( m | λ ) = p ( µ ), where µ = m − λ . We can refer to thisproperty as shift invariance [12]. The performance of thewhole detection process is given only by the form of p ( µ )that depends just on ρ , ∆ , and G . More specifically p ( µ ) is fixed by the coherence properties of ρ and ∆ with respect to G , as we shall see clearly below.Before proceeding let us present a necessary conditionthat ∆ must satisfy so that the shift-invariant ruler∆( m ) = D ( m )∆ D † ( m ) provides a bona fide measure-ment. Given the fomr of ∆( m ), the necessary and suf-ficient conditions for the reality, nonnegativity, and nor-malization of p ( µ ) are just nonnegativity of ∆ and res-olution of identity, this is Z dm ∆( m ) = I, (2.5)where I is the identity. To show the consequences of con-dition (2.5) let us use the basis of eigenvectors of G , thisis G | g i = g | g i , where g will be assumed to be continuousand nondegenerate without loss of generality, with Z dg | g ih g | = I. (2.6)Then Z dm ∆( m ) = Z dm Z dg Z dg ′ | g ih g | ∆( m ) | g ′ ih g ′ | . (2.7)Using Eq. (2.1) and D ( m ) | g i = exp( − igm ) | g i , Z dm ∆( m ) = Z dm Z dg Z dg ′ e im ( g − g ′ ) | g ih g | ∆ | g ′ ih g ′ | , (2.8) so that performing the m integration Z dm ∆( m ) = 2 π Z dg | g ih g | ∆ | g ih g | . (2.9)The necessary and sufficient conditions that ∆ must sat-isfy to provide a legitimate quantum ruler ∆( m ) are∆ > , h g | ∆ | g i = 12 π , (2.10)which implies h g | ∆( m ) | g i = 12 π . (2.11)This condition is quite interesting since it means that∆( m ) are nontrivial just because of their nondiagonalterms in the generator basis. These are their coherenceterms with respect to G , in the way coherence is under-stood in modern quantum mechanics. And to say coher-ence is to say nonclassicality. As a further consequence,condition (2.10) implies that the variance of G in ∆( m )diverges. III. QUANTUM WIENER-KINTCHINETHEOREM
Let us now address the form of p ( µ ) given its relationto ρ , ∆ , and G in Eq. (2.4). We can proceed from Eq.(2.4) making use of the G basis to get p ( µ ) = Z dg Z dg ′ h g | ρ | g ′ ih g ′ | ∆ | g i e i ( g − g ′ ) µ . (3.1)In order to highlight the relation between p ( µ ) and co-herence, let us formally perform the change of variables g ′ = g + τ , to get p ( µ ) = Z dτ Γ( τ ) e − iτµ , (3.2)where Γ( τ ) = Z ′ dg h g | ρ | g + τ ih g + τ | ∆ | g i , (3.3)and by the prime we mean that the range of integrationon g may depend on τ . Note that this relation naturallyrespects the conditions that p ( µ ) must satisfy as a proba-bility density, this is to be real, positive, and normalized,so we have Γ(0) = 1 / (2 π ) as well as Γ ∗ ( τ ) = Γ( − τ ).We see that Eqs. (3.2) and (3.3) provide a quantumexact replica mutatis mutandis of a celebrated theoremof classical coherence optics, the Wiener-Kintchine theo-rem [5, 23], that establishes that the spectral density andthe coherence function are a Fourier transform pair. Inour case the statistics p ( µ ) plays the role of the spectraldensity while Γ( τ ) plays the role of coherence function.We might better say that Γ( τ ) is a detection-processcoherence function made by the product of the individualprobe and ruler coherence functions h g | ρ | g + τ i , h g | ∆ | g + τ i , (3.4)respecting the fruitful equivalence between the matrix el-ements of the density matrix and a classical-optics coher-ence function. To ensure this coherence link let us notethat the full dependence of Γ( τ ) on τ relies just on thecoherence terms, this is the nondiagonal terms of probeand ruler on the basis G , which is precisely the coherencebasis that matters regarding metrology.As a rather relevant result of this coherence analysis,we highlight that the detection-process coherence func-tion is fully symmetrical on the probe and the ruler, sothe probe and ruler coherences contribute equally to theresolution.A final and interesting remark of this approach is thatthe coherence with respect to the observable G is notthe same that coherence with respect to the observable G , even though both G and G can be diagonal in thesame basis, so that the traditional resource theories willpredict the same coherence. This is further illustrated inAppendix A. A. Resolution and coherence time
The above theorem leads us to a natural relation be-tween signal-detection resolution ∆ λ , understood as thewidth of the spectral-like function p ( µ ), and coherencetime τ c , understood as the width of the coherence-likefunction Γ( τ ).To suitably establish their relation several approachesmight be followed. Here we will follow the one presentedin Ref. [13] as particularly simple and insightful, whichcan be readily derived from the Parceval’s relation2 π Z dτ | Γ( τ ) | = Z dµp ( µ ) . (3.5)Based on this we define the following measures of coher-ence and signal uncertainty, both relying in some versionof Renyi entropies [15]. Regarding coherence let us con-sider τ c = Z dτ | γ ( τ ) | , (3.6)where γ ( τ ) is the corresponding degree of coherence asthe properly normalized coherence function γ ( τ ) = Γ( τ )Γ(0) = 2 π Γ( τ ) . (3.7) Concerning resolution, we define the signal uncertainty∆ λ as ∆ λ = 12 √ π R dµp ( µ ) . (3.8)Combining both we get the resolution-coherence relation τ c ∆ λ = √ π. (3.9)So the signal uncertainty is inversely proportional to thecoherence time. Note that the µ integration of p ( µ ) isnormalized to one while the τ integration of Γ( τ ) is not,so the two width measures look so different although theyaddress the same idea. B. Gaussian model
Let us elaborate further this relation with a particularGaussian form for probe and ruler, say ψ ( g ) = 1 p ∆ G √ π e − ( g − g ) / (4∆ G ) e ik g . (3.10)where we assume the probe to be in a pure state ρ = | ψ ih ψ | , being ψ ( g ) = h g | ψ i , and h g | ∆ | g ′ i = 12 π e − ∆ Φ M ( g − g ′ ) / , (3.11)where as before we refer as Φ to the observable conjugateto G . Resorting once again to classical optics, quantumGaussian states parallels Gaussian Schell-model sources[5, 17]. In these terms, the factor ∆Φ M actually expressescoherence in the degree of freedom expressed by G .With all this we get thatΓ( τ ) = 12 π e − (∆ Φ M +∆ Φ S ) τ / , (3.12)being ∆ Φ S = 14∆ G . (3.13)This leads to a coherence time τ c = π ∆ Φ M + ∆ Φ S , (3.14)and then, finally, the resolution becomes:∆ λ = ∆ Φ M + ∆ Φ S . (3.15) C. Fisher Information
The results of this approach can be compared andcomplemented by the resolution lower limit given by theCram´er–Rao bound [8, 9]∆ λ = 1 F , (3.16)with Fisher information F = Z dm p ( m | λ ) (cid:20) ∂p ( m | λ ) ∂λ (cid:21) = Z dµ p ( µ ) (cid:20) ∂p ( µ ) ∂µ (cid:21) . (3.17)Likewise, the Fisher information is bounded by aboveby the quantum Fisher information [10]. For pure states,the quantum Fisher information becomes proportional tothe variance on the probe state of the generator of thesignal-dependent transformation, F ≤ F Q = 4∆ G. (3.18)As well as the previous one, this method also takes intoaccount the resources conveyed by the detector. This en-ables us to distinguish between ideal, unrealistic measure-ments and more realistic measurements, where by idealwe mean that it does not contribute to the signal uncer-tainty, say ∆Φ M = 0. This may be crucial for furtheroptimization of the resources involved. IV. LINEAR METROLOGY
The above general analysis fits perfectly well to sig-nal detection based on the Heisenberg-Weyl group oftransformations. To this end let us consider the one-dimensional motion of a particle with position- andmomentum-like operators satisfying the commutation re-lation [
X, P ] = i . This is equally valid for a one-modefield where X and P are the field quadratures. Morespecifically, let it be G = P so we are detecting positionshifts.We will consider pure Gaussian states both for theprobe and for the measurement. For the probe state | ψ i we have in the position representation ψ ( x ) = 1 p ∆ X S √ π e − ip x e − ( x − x ) / (4∆ X S ) , (4.1)where as usual ψ ( x ) = h x | ψ i being | x i the eigenstatesof position operator X , say X | x i = x | x i .Concerning the measurement we consider statisticsgiven by projection on squeezed coherent states | ϕ m,k i as being the displacement of a squeezed vacuum. In theposition representation their wavefunction is ϕ m,k ( x ) = 1 p ∆ X M √ π e − ikx e − ( x − m ) / (4∆ X M ) , (4.2)being ϕ m,k ( x ) = h x | ϕ m,k i . Note that we have the shift-invariant property granted in particular in the m vari-ables given that | ϕ m,k i = D ( m ) | ϕ ,k i . Since all occurson the x -domain we can simplify matters if we get ride of the k and p dependences in the usual way of lookingfor the marginal for the m outcome, so our POVM is∆( m ) = 12 π Z dk | ϕ m,k ih ϕ m,k | , (4.3)leading to a fully shift-invariant detector model ∆( m ) = D ( m )∆ D † ( m ) with∆ = 12 π Z dk | ϕ ,k ih ϕ ,k | , (4.4)leading to∆ = 1 √ π ∆ X M Z dxe − x / (2∆ X M ) | x ih x | . (4.5)In the P representation we get h p | ∆ | p ′ i = 12 π e − ∆ X M ( p − p ′ ) / , (4.6)in agreement conditions Eq. (2.10) and the Gaussianmodel (3.11).With all this we get as suitable particular examples ofEqs. (3.12), (3.14), and (3.15)Γ( τ ) = 12 π e − (∆ X M +∆ X S ) τ / , (4.7)leading to a coherence time τ c = π ∆ X M + ∆ X S , (4.8)and finally a signal uncertainty∆ λ = ∆ X M + ∆ X S . (4.9) A. Cram´er–Rao bound
It is worth noting that ∆ λ coincides exactly with theCram´er–Rao bound∆ λ = 1 F P = 1 X M +∆ X S . (4.10)whit F P being the Fisher information (3.17). This mightbe expected since we can take the outcomes m as a suit-able estimator so that a Gaussian shift-invariant condi-tional distribution implies that the estimator is efficient[11].Summarizing, the larger the squeezing the lesser∆ X M,S , and thus the larger the coherence and the largerthe resolution. This is a natural and intuitive relation,that nevertheless is not satisfied by other approaches tocoherence where larger squeezing means lesser coherenceas shown for example in Ref. [18]. Actually, Fisher infor-mation has already been proposed by itself as a suitablemeasure of coherence in Refs. [19].
B. Optimization
This resolution (4.9) can be further compared to theCram´er–Rao bound under an ideal X measurement withprojection-valued measure ∆( m ) = | x = m ih x = m | , forthe same Gaussian probe state (4.1). In such a case weobtain ∆ λ = 1 F P = ∆ X S , (4.11)with F P = 1∆ X S = 4∆ P S , (4.12)where the last equality for F P shows that it is actuallythe quantum Fisher information (3.18).Comparing Eqs. (4.9) and (4.11) we can see the nat-ural result that the ideal case is retrieved in the limit∆ X M →
0. But it must be noted that this means aninfinite amount of coherence or squeezing resources de-voted to the measurement. This holds whatever the wayresources are counted, this means either infinite coher-ence time, infinite nonclassicality, infinite squeezing, oreven infinite energy. For example, in quantum optics theprojection on the quadrature eigenstates | x i is carriedout in an hodomdyne detection scheme only in the limitof a local oscillator much more intense that the systemstate being measured. This essentially means that in thisunrealistic limit all resources are devoted to the measure-ment, which is a fact normally no taken into account.Therefore, it is worth examining the optimization inthe case of finite resource including the resources em-ployed in the measurement.It is clear that the resource determining the resolutionis the coherence expressed by the coherence times X M and X S . So let us consider as resource some fixedamount of coherence split between probe and ruler sothat the following quantity is held constant1∆ X S + 1∆ X M = constant = C. (4.13)It can be seen that the minimum of the signal uncer-tainty in Eq.(4.9) when C is held constant holds whenthe split of the coherence is balanced between probe andruler, this is X M = X S , giving a signal uncertaintyjust 2 times the ideal case in Eq. (4.11), this is∆ λ = 2∆ X S . (4.14)We would like to point out that there is no relationwhatsoever between resolution ∆ λ and energy, since ac-tually we have never specified the physical apparatus em-bodying the probe and ruler, i. e., this might be a freeparticle, or an harmonic oscillator, or anything else, sothe Hamiltonian H might be anything without alteringa bit of the conclusions. This analysis shows that whatreally matters is the internal coherence structure of bothprobe and measurement. V. PHASE SHIFTS
Let us consider one of the most useful and studiedgenerators. This is the free Hamiltonian of the harmonicoscillator, the number operator NG = N = 12 (cid:0) P + X (cid:1) , (5.1)which includes as the most relevant examples the freeevolution of single mode fields. By the way, this is a bal-anced combination of the two basic nonlinear generators,we will come again to this later. In any case, the phaseshift generated by N is the fundamental basis of interfer-ence, which the most powerful detector as demonstratedin the detection of gravitational waves [21]. A. Quantum ruler
There is a possibility to directly translate to this casethe very general analysis made above in Sec. 4. To bemetrologically useful the probe states experiencing phaseshifts must have a very large mean number of photons,which allows the useful approximation of the discretespectrum of N by a continuous one, extending its domainto the entire real axis as a good simplifying assumption.Similarly regarding the domain of variation for the phase.In this case, there is a readily physical picture of theobservable Φ conjugate to G , This is the quantum-opticalphase observable [22]. For a single-mode field Φ can bewell described by the positive operator valued measure∆( φ ) = | φ ih φ | , (5.2)where | φ i are the nonorthogonal, unnormalized Susskind-Glogower phase states | φ i = 1 √ π ∞ X n =0 e − iφn | n i , (5.3)being | n i the photon-number states as the eigenstates ofthe number operator N , this is N | n i = n | n i . There isclearly shift invariance, since∆( φ ) = D ( φ )∆ D † ( φ ) , (5.4)with D ( φ ) = e − iφN , ∆ = ∆( φ = 0) . (5.5)This represents the case of an unrealistic phase measure-ment. To deal with a more realistic one we just replace∆ by∆ = 1 √ π ∆Φ M Z dφe − φ / (2∆ Φ M ) | φ ih φ | , (5.6)leading in the number representation to h n | ∆ | n ′ i = 12 π e − ∆ Φ M ( n − n ′ ) / , (5.7)and we have used the above-mentioned approximationfor the variable n . Furthermore, we can consider thatthe probe may be described by a Gaussian in the numberbasis as ψ ( n ) ≃ p ∆ N S √ π e − ( n − ¯ n ) / (4∆ N S ) . (5.8)With all this we get thatΓ( τ ) ≃ π e − (∆ Φ M +∆ Φ S ) τ / , (5.9)where ∆ Φ S ≃ N S . (5.10)This leads to a coherence time τ c = π ∆ Φ M + ∆ Φ S , (5.11)and then finally ∆ λ = ∆ Φ M + ∆ Φ S , (5.12) which clearly reproduces the structure of preceding re-sults. B. Cram´er–Rao bound
For the same settings, this is, a Gaussian probe state(5.8) and a unrealistic, Gaussian, measurement (5.6), theobtained uncertainty (5.12) coincides exactly with theCram´er–Rao bound calculated by (3.16) and (3.17),∆ λ = ∆ Φ M + ∆ Φ S . (5.13)In the case of unrealistic phase measurement, ∆( φ ) = | φ ih φ | , the only contribution that remains is the uncer-tainty of the probe state, ∆ λ = ∆ Φ S . As in the linearcase, this uncertainty coincides with the inverse of thequantum Fisher information. C. Coherent-squeezed scheme
It can be interesting to analyze the performance of thescheme introduced in the preceding section in terms ofthe more accessible settings regarding the probe stateand the linear ruler in Eqs. (4.1) and (4.2), respectively.In this case, the Fisher information can be computed using the good transformation properties of Wigner function.At λ = 0, which may properly account for the case of small enough signal values, it leads to F N = (cid:0) ∆ X S − ∆ P S (cid:1) (∆ X S + ∆ X M ) (∆ P S + ∆ P M ) + x ∆ P S + ∆ P M + p ∆ X S + ∆ X M . (5.14)The terms depending on the displacements x and p reproduce the structure that we found in the following sectiondevoted to the nonlinear case G = P . D. Resolution as phase uncertainty
Let us show that the Fisher information (5.14) can befully expressed in terms of phase uncertainty. We findthis quit suggestive since phase fluctuations are a keytool to understand coherence in the classical domain.To this end we can join ∆ X M to ∆ X S as being addi-tional effective uncertainty caused by some kind of blurryorigin as ∆ ˜ X S = ∆ X S + ∆ X M . (5.15)So the combination ∆ ˜ P S /x is the phase uncertainty fora state centered at x with a ∆ ˜ P S uncertainty along Y .Similarly for the term ∆ ˜ X S /p . So the last two terms are determined for phase uncertainty related to the dis-placement term in Eq. (5.14).On the other hand, the first term in Eq. (5.14) is in-dependent of the displacements x and p , so it can besuitably understood as phase uncertainty for an squeezedvacuum. We can address this quadrature-based phaseuncertainty for the vacuum in terms of its Wigner func-tion for the squeezed vacuum W S ( x, p ) where the coordi-nates x, p refer to the blurry variables ˜ X S , ˜ P S includingthe fluctuations added by the detection process. Afterchanging to polar coordinates x = r cos φ , p = r sin φ and integrating on r to get a phase distribution W S ( φ ) W S ( φ ) ∝
11 + (cid:16) ˜ X S − ˜ P S (cid:17) sin φ . (5.16)This phase distribution very much recalls the resolutionof a Fabry-Perot interferometer [23] suggesting a phaseuncertainty of the form∆ φ ≃ (cid:12)(cid:12)(cid:12) ˜ X S − ˜ P S (cid:12)(cid:12)(cid:12) = ∆ ˜ X S ∆ ˜ P S (cid:12)(cid:12)(cid:12) ∆ ˜ X S − ∆ ˜ P S (cid:12)(cid:12)(cid:12) . (5.17)So, grosso modo we have that the Fisher informationfor phase shifts is fully expressible in terms of phase un-certainties as F N ∝ φ + 1∆ φ x + 1∆ φ p . (5.18) E. Non-Gaussian scenario
To conclude, we employ the phase shift detection totest our quantum Wiener-Kintchine theorem when ap-plied to non-Gaussian states. To this end, we consider anideal phase measurement (5.2) over a normalizable ver-sion of the Susskind-Glogower phase states as the probestate, | ξ i = p − | ξ | ∞ X n =0 ξ n | n i , (5.19)specially interesting when | ξ | → λ = π (cid:18) − | ξ | | ξ | (cid:19) . (5.20)We compare this result with the arisen from calculatingthe Fisher information in (3.17), which conduces to∆ λ = (1 − | ξ | ) | ξ | . (5.21)It is worth noting the similarity between the findingsof both methods, regarding that we are working with | ξ | →
1. In this limit, the quantum Wiener–Kintchinetheorem gives a resolution of π (1 −| ξ | ) and the Cram´er–Rao bound of 2(1 − | ξ | ) .This example shows the generality of the theorem de-veloped in Sec. III beyond the more developed Gaussianmodel. VI. NONLINEAR METROLOGY
By nonlinear metrology we refer to generators G be-yond the Heisenberg-Weyl group of transformations, forexample let G = P . In this case, the rather accessible settings already usedabove for the linear metrology, with Gaussian forms forthe probe and the measurement, do not satisfy the in-variance condition in (2.10). The general result obtainedin (3.9), still valid, would need a different physical imple-mentation which respects the invariance condition. A. Cram´er–Rao bound
Nevertheless, in order to extend our analysis of therelation between resolution and coherence to the nonlin-ear case, we make the ansatz supported by the preced-ing examples that the Fisher information provides a toolcompatible with the approach presented above. Thus,we analyze the performance of nonlinear detection viathe Fisher Information regarding the probe state and thelinear ruler in the form specified in Eqs. (4.1) and (4.2),respectively. This is we consider probe and ruler beingjust plain coherent-squeezed states and we relate the res-olution to the squeezing as done in the linear case . Wemust take into account that now the k outcomes are nottrivial and contribute to the statistics.Then, we estimate the resolution via the Cram´er–Raobound ∆ λ = 1 F P . (6.1)The Fisher information evaluated at λ = 0, becomes F P = ∆ P S ∆ P M F P + 4 p F P , (6.2)where ∆ P S,M represent the uncertainties of the operator P in the probe and measurement states∆ P S,M = 12∆ X S,M , (6.3)and F P is the Fisher information corresponding to thelinear case, G = P , F P = 1∆ X + ∆ X M . (6.4)Let us compare again the obtained Fisher information(6.2) with the Fisher information in the case of an unre-alistic measurement of the observable X after the actionof the nonlinear signal-induced transformation generatedby G = P . At λ = 0 we obtain F P = 16 p ∆ P S , (6.5)which coincides with the limit ∆ X M → B. Linear and nonlinear terms
There are two well differentiated contributions to theFisher information F P in Eq. (6.2). This splitting canbe easily understood is we transfer the displacement p from the probe state to the transformation, this is P → ( P + p ) = P + 2 p P + p , (6.6)where the transformation generated by the right-handside observable is acting on a probe with vanishing meanmomentum. The constant term p is trivial regardingtransformations since it produces no effect. So we getthe combination of a nonlinear transformation P and alinear transformation 2 p P acting on a probe with van-ishing mean momentum. In the linear part 2 p P we canobserve that the momentum displacement p acts ampli-fying the signal value λ by a factor 2 p . Since we alreadyknow well about the linear part let us develop further thefirst nonlinear term in Eq. (6.2) considering p = 0 F P = ∆ P S ∆ P M F P , (6.7)expressing it in terms of X variances using ∆ X ∆ P = 1 / F P = ∆ X M / ∆ X S (∆ X M + ∆ X S ) , (6.8)and finally F P = 2∆ X M ∆ X S (∆ X M + ∆ X S ) QF P (6.9)where QF P is the quantum Fisher information of theprobe QF P = 4∆ G = 12 (∆ X S ) . (6.10) C. Optimization nonlinear term
The first factor in Eq. (6.9) depends symmetrically onthe probe and measurement as in the linear case. Fora fixed joint amount of coherence (4.13), the optimumvalue of this nonlinear term is obtained when1∆ X S = 3 1∆ X M , (6.11)leading for this contribution to F P = QF P . VII. CONCLUSIONS
We have developed a theory of quantum coherencesuited to be applied to quantum metrology. This formu-lation shows via a simple model that coherence is the true resource behind resolution, as suitably expressed by aquantum version of the Wiener–Kintchine theorem. Thisis also confirmed by the more common Fisher informationanalysis.One of the virtues of this formalism is that it showsthat both the apparatus and probe coherence propertiescontribute equally to the detection performance. This isa valuable result that acknowledges that system and de-tector are inextricably mingled to produce the observedstatistics as required by basic quantum postulates suchas the very Born’s rule.
Appendix A: Coherence relative to G and G aredifferent As commented within the body of the manuscript, areally interesting property of the theory introduced inthis work is that the coherence with respect to the ob-servable G is not the same that coherence with respect tothe observable G , even though both G and G can be di-agonal in the same basis, so that the traditional resourcetheories will predict the same coherence.To show this let us consider G = P and G = P = G within a model of the most simple case of a pureGaussian state for the probe ψ ( p ) = 1 p ∆ P √ π e − ( p − p ) / (4∆ P ) . (A1)Let us focus on a simplified version of the coherence func-tion just missing the apparatus part for simplicity. Forthe case G = P we getΓ ( τ ) = Z ′ dp h p | ρ | p + τ i = e − τ / (8∆ P ) . (A2)leading to a coherence time τ c ∝ ∆ P independent of p .In the case G = P = G we get, for p = 0Γ ( τ ) = Z ′ dp h p | ρ | p p + τ i = e −| τ | / (4∆ P ) , (A3)leading to a coherence time τ c ∝ ∆ P . Moreover it canbe seen numerically that when p = 0 the result clearlydepends on p .It might be argued that the main difference betweencoherence function (A2) and (A3) is a matter of nomen-clature and that the results are equivalent provided wereplace τ by τ in Eq. (A3). However, this is not quiteso, since in such a case we should also replace τ by by τ in the Wiener-Kintchine theorem (3.2), so that the finalstatistics is the same using either τ or τ . In other words,the τ differences between Eqs. (A2) and (A3) are fullymeaningful in the sense that they express the differentsignal resolutions provided by the linear and nonlinearschemes. Acknowledgments.-
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