Single- versus two-parameter Fisher information in quantum interferometry
aa r X i v : . [ qu a n t - ph ] J u l Single- versus two-parameter Fisher information in quantum interferometry
Stefan Ataman
Extreme Light Infrastructure - Nuclear Physics (ELI-NP),‘Horia Hulubei’ National R&D Institute for Physics and Nuclear Engineering (IFIN-HH),30 Reactorului Street, 077125 M˘agurele, jud. Ilfov, Romania ∗ (Dated: July 8, 2020)In this paper we reconsider the single parameter quantum Fisher information (QFI) and compareit with the two-parameter one. We find simple relations connecting the single parameter QFI (bothin the asymmetric and symmetric phase shift cases) to the two parameter Fisher matrix coefficients.Following some clarifications about the role of an external phase [Phys. Rev. A , 011801(R)(2012)], the single-parameter QFI and its over-optimistic predictions have been disregarded in theliterature. We show in this paper that both the single- and two-parameter QFI have physicalmeaning and their predicted quantum Cram´er-Rao bounds are often attainable with the appropriateexperimental setup. Moreover, we give practical situations of interest in quantum metrology, wherethe phase sensitivities of a number of input states approach the quantum Cram´er-Rao bound inducedby the single-parameter QFI, outperforming the two-parameter QFI. I. INTRODUCTION
Interferometric phase sensitivity is a research topic ofinterest for a number of rapidly growing scientific fields,among which we can single out gravitational wave as-tronomy [1–8] and quantum technologies [9–12].With the advent of non-classical states of light [13–16],the classical SNL (shot-noise limit) [17] has been shownto be improvable [18], prediction confirmed by experi-ments [19–21].Theoretical bounds for the interferometric phase sen-sitivity became possible due to the quantum Fisher in-formation (QFI) and its associated quantum Cram´er-Raobound (QCRB) [22–27]. These bounds, besides their the-oretical interest, are extremely useful in evaluating theoptimality of realistic detection schemes.Jarzyna & Demkowicz-Dobrza´nski [28] showed in aconvincing manner that using the single-parameter QFIconstantly yields over-optimistic results. As pointed out,the solution to avoid counting resources that are actuallyunavailable is to phase-average the input state [28–30] oruse the two-parameter QFI [31–36].One reason that the single parameter QFI might beconsidered over-optimistic or artificial is that usual de-tection schemes cannot go beyond the QCRB given by atwo-parameter QFI approach [31, 33, 37]. Even the bal-anced homodyne detection, although having access to anexternal phase reference, cannot exceed this limit if theinterferometer is balanced [36, 37].As discussed in previous works [28, 30], actual phasemeasurement scenarios are modeled with a single phaseshift for some applications [38, 39], while others requiretwo phase shifts [1]. Thus, we consider both these sce-narios in this work.Gaussian input states are a popular choice due to boththeir properties and to technical advancements in their ∗ [email protected] preparation [40]. Among them we can cite the coherentplus squeezed vacuum input state [18, 41, 42], a popularchoice also due to its use in gravitational wave detection[3–7]. The squeezed coherent plus squeezed vacuum in-put state [43] has been shown to bring a gain in phasesensitivity due to the second squeezer [34, 36]. This gain,however, becomes marginal in the experimentally inter-esting scenario of high input coherent power and limitedsqueezing factors. In this paper, we will show how toovercome this limitation using an unbalanced interfero-meter and an external phase reference. We also notethat, on the experimental side, squeezing a laser sourcehas been recently demonstrated [6].Although most authors employ balanced (50/50) in-terferometers [24, 37, 41, 42, 44], a number of works ad-dressed the unbalanced scenarios, too [28, 34, 35]. Someinteresting results emerged, for example in the case ofdouble coherent input [34].Reference [28] gave reasons not to use a single-parameter QFI. In this paper we take exactly the op-posite route: we find scenarios where using a single-parameter QFI is interesting. Moreover, we find detec-tion schemes that are actually able to reach the QCRBpredicted by the single-parameter QFI. However, in orderto do so, we need to employ an unbalanced interferome-ter.The input phase matching conditions (PMC) and theireffect on performance have been discussed in the litera-ture [28, 34, 36, 45]. Although in some works all inputphases are set to zero [28, 43, 45], this is not always anoptimal choice [34, 36]. In this paper we will show thatthe optimal PMCs change not only in function of theinput state, but also with the type of QFI used.In this work we focus on two detection schemes. Thedifference-intensity detection scheme is often consideredin the literature [24, 33, 34, 36, 37] and it is a good ex-ample of a detection method not having access to an ex-ternal phase reference. We thus expect its performanceto be limited by the two-parameter QFI. The homodynedetection technique [36, 37, 46–48] is the quintessentialexample of a detector having access to an external phasereference. We will show that under the right conditions,it is able outperform the QCRB implied by the two-parameter QFI, approaching the one corresponding tothe single parameter QFI.This paper is structured as follows. In Section IIwe introduce some conventions and describe the two-parameter QFI approach. In Section III we discuss thesingle-parameter QFI with an asymmetric phase shiftwhile in Section IV we discuss the same problem in thesymmetric phase shift scenario. In Section V we givethe complete expression for these QFIs for an importantclass of input states, namely the Gaussian states. Therealistic detection schemes to be considered in this paperare described in Section VI. The performance of theseschemes with some Gaussian input states is detailed anddiscussed in Section VII. The results are discussed andsome assertions from the literature commented in SectionVIII. The paper closes conclusions in Section IX. II. TWO PARAMETER QUANTUM FISHERINFORMATION
Throughout this work we assume no losses and ourinput is limited to a pure state, thus we do not need touse the Symmetric Logarithmic Derivative [22, 24, 26].We also assume no entanglement between the two inputports. This is a rather standard assumption in papersdiscussing Gaussian input states [31–34, 36, 37].We first consider the general case where each arm ofthe interferometer contains a phase-shift ( ϕ and, respec-tively, ϕ , see Fig. 1). BS denotes the beam splitter. Theestimation is treated as a general two parameter problem[28, 31–33]. We define the 2 × F = (cid:20) F ss F sd F ds F dd (cid:21) (1)where the coefficients are defined by F ij = 4 ℜ{h ∂ i ψ | ∂ j ψ i − h ∂ i ψ | ψ ih ψ | ∂ j ψ i} (2)with i, j ∈ { s, d } , ϕ s/d = ϕ ± ϕ and ℜ denotes the realpart. We also denote | ∂ s/d ψ i = ∂ | ψ i /∂ϕ s/d and we have | ψ i = e − i ˆ n − ˆ n ϕ d e − i ˆ n n ϕ s | ψ i where ˆ n m = ˆ a † m ˆ a m de-notes the number operator for port (mode) m . We em-ploy the usual annihilation (creation) operators ˆ a m (ˆ a † m )obeying the commutation relations [ˆ a m , ˆ a † n ] = δ mn with m, n labeling spatial modes.From the Fisher matrix (1) we arrive at a QCRBmatrix inequality [31] out of which we retain only thedifference-difference phase estimator,(∆ ϕ d ) ≥ ( F − ) dd (3)and since the matrix element ( F − ) dd will appear repeat-edly we define the two-parameter QFI, F (2 p ) := 1( F − ) dd = F dd − F sd F ds F ss (4) BS M reference beam FIG. 1. The configuration for the case study with two inde-pendent phase shifts, ϕ and ϕ . The beam splitter BS isassumed to have a variable transmission coefficient, T . thus saturating inequality (3) implies the two-parameterQCRB, ∆ ϕ (2 p ) QCRB = 1 √F (2 p ) . (5)Since the QFI is additive (both for the single- and twoparameter cases) [22, 24], for N repeated experiments wehave the scaling ∆ ϕ (2 p ) QCRB = √ N F (2 p ) . For simplicity,throughout this paper we set N = 1.For the calculation of the Fisher matrix elements weneed the field operator transformations, (cid:26) ˆ a = R ˆ a + T ˆ a ˆ a = T ˆ a + R ˆ a (6)where T ( R ) denotes the transmission (reflection) coef-ficient of the beam splitter ( BS in Fig. 1). We have | T | + | R | = 1 and T R ∗ + T ∗ R = 0 [49]. Since thelast relation implies ( T ∗ R ) = −| T R | , a sign conventionhas to be made (i. e. T ∗ R = ± i | T R | ). Without loss ofgenerality, throughout this paper we use the convention iT ∗ R = −| T R | and for the particular case of balancedBS we consider T = 1 / √ R = i/ √ F ss can now be computed andyields F ss = ∆ ˆ n + ∆ ˆ n . (7)By the variance ∆ ˆ n k we denote h ψ | ˆ n k | ψ i − h ψ | ˆ n k | ψ i and the standard deviation is ∆ˆ n k = √ ∆ ˆ n k . F dd iscomputed and the result is given in equation (A2). Thelast term we need is F sd since F sd = F ds [31] and theresult is given in equation (A3).In the balanced case F ss remains unchanged while thedifference-difference Fisher matrix element F dd becomes F dd = h ˆ n i + h ˆ n i + 2 (cid:0) h ˆ n ih ˆ n i − |h ˆ a i| |h ˆ a i| (cid:1) − ℜ (cid:16) h ˆ a ih (ˆ a † ) i − h ˆ a i h ˆ a † i (cid:17) (8)and F sd reduces to F sd = 2 ℑ (cid:16) h ˆ a ih ˆ a † i + ( h ˆ n ˆ a i − h ˆ n ih ˆ a i ) h ˆ a † i + h ˆ a i (cid:16) h ˆ a † ˆ n i − h ˆ a † ih ˆ n i (cid:17)(cid:17) (9)where ℑ denotes the imaginary part [36]. Throughoutthis work we assume that the input port 1 is never in thevacuum state, i. e. h ˆ n i 6 = 0.Interferometric phase sensitivity is based on the phasedifference induced between the two arms of an interfero-meter. In most cases the optimal sensitivity is obtainedin the balanced case [31, 32], but exceptions have beenshown to exist [30, 34]. Thus, if we stray away from thebalanced case until the extreme | T | → | T | → | T | → F ss from equation (7) remainsunchained and applying the limit | T | → F dd = ∆ ˆ n + ∆ ˆ n . The same constraintapplied to F sd from equation (A3) yields F sd = ∆ ˆ n − ∆ ˆ n . (10)If F sd = 0 i. e. if ∆ ˆ n = ∆ ˆ n then equation (4) implies F (2 p ) = 2∆ ˆ n . (11)If F sd = 0, the two-parameter difference-difference equiv-alent QFI becomes F (2 p ) = 4 ∆ ˆ n ∆ ˆ n ∆ ˆ n + ∆ ˆ n (12)and somehow surprisingly F (2 p ) = 0 except when the in-put state 0 is in the vacuum state. However, one shouldnot overlook the fact that although the two-parameterQFI guarantees not to consider resources obtainable viaan external phase reference, the input being in a purestate, it implies a fixed phase relation between the quan-tum states from ports 0 and 1. Since a well chosen de-tection scheme can take advantage of this fact, equation(12) should be less surprising. III. SINGLE PARAMETER QUANTUM FISHERINFORMATION WITH AN ASYMMETRICPHASE SHIFT
In the single-parameter case (see Fig. 2), the QFI issimply [22, 24, 26] F ( i ) = 4 (cid:0) h ∂ ϕ ψ | ∂ ϕ ψ i − |h ∂ ϕ ψ | ψ i| (cid:1) , (13)and we use here the notations from reference [28]. Weassume a single phase shift in the output 3 of BS , i. e.we model it as ˆ U ( ϕ ) = e − iϕ ˆ n , thus the QFI is formallygiven by F ( i ) = 4∆ ˆ n (14) BS M reference beam FIG. 2. The configuration for the case study employing asingle phase shift. The beam splitter BS is assumed to havea variable transmission coefficient, T . and it implies the (single parameter) QCRB∆ ϕ ( i ) QCRB = 1 √F ( i ) . (15)The calculations for F ( i ) are detailed in Appendix B andthe final result with respect to the input parameters isgiven in equation (B1). Comparing equation (B1) withthe Fisher matrix elements from the previous section wenote that the single-parameter Fisher information can beexpressed as a function of the Fisher matrix elements andwe have F ( i ) = F ss + F dd − F sd . (16)From the definition of the two-parameter QFI (4) andthe above equation we can immediately prove that F ( i ) ≥ F (2 p ) (17)with equality only if F ss = F sd . In the balanced case theQFI F ( i ) simplifies to the expression given by equation(B2). In the limit case | T | →
1, the QFI from equation(B1) reduces to F ( i ) = 4∆ ˆ n (18)a result that might look surprising, since all terms relatedto input port 0 are missing. However, in this degeneratecase only input port 1, can “reach” the phase shift ϕ ,hence the result. IV. SINGLE PARAMETER QUANTUM FISHERINFORMATION WITH TWO SYMMETRICPHASE SHIFTS
In this last scenario (see Fig. 3), we assume a dis-tributed phase shift of ϕ/ − ϕ/
2) in the output port3 (2) of the beam splitter BS , i. e. we model it asˆ U ( ϕ ) = e − i ϕ ˆ n + i ϕ ˆ n , thus the QFI is given by F ( ii ) = ∆ ˆ n + ∆ ˆ n . (19) BS M reference beam FIG. 3. The configuration for the case study with symmetric,(anti)correlated phase shifts, ± ϕ/
2. The beam splitter BS is assumed to have a variable transmission coefficient, T . The calculations are detailed in Appendix C and the finalform of F ( ii ) is given in equation (C1). This QFI impliesthe QCRB ∆ ϕ ( ii ) QCRB = 1 √F ( ii ) . (20)In this scenario, too, we find a simple relation connect-ing F ( ii ) to the two-parameter Fisher matrix elements,namely F ( ii ) = F ss F dd F ( ii ) and F (2 p ) .In the balanced case F ( ii ) simplifies to the expressiongiven by equation (C2). In the limit | T | → F ( ii ) = ∆ ˆ n + ∆ ˆ n . (22) V. GAUSSIAN INPUT STATES AND THEIRRESPECTIVE QFI
In this section we discuss the three previously intro-duced QFI metrics (i. e. F (2 p ) , F ( i ) and F ( ii ) ) with anumber of Gaussian input states. A. Single coherent input
In this simple scenario we consider the input state | ψ in i = | α i = ˆ D ( α ) | i (23)where the displacement or Glauber operator [49–51] fora port k is defined byˆ D k ( α ) = e α ˆ a † k − α ∗ ˆ a k . (24) The first Fisher matrix element is F ss = | α | and fromequation (A2) we get F dd = (cid:0) | T | − | R | (cid:1) | α | + 4 | T R | | α | = | α | (25)Finally, equation (A3) gives F sd = − (cid:0) | T | − | R | (cid:1) | α | and from equation (4) we obtain the two-parameter QFI, F (2 p ) = 4 | T R | | α | (26)and it implies the QCRB∆ ϕ (2 p ) QCRB = 1 √F (2 p ) = 12 | T R || α | (27)yielding in the balanced case the well-known shot-noiselimit ∆ ϕ (2 p ) QCRB = 1 / | α | [24, 31, 37]. This limit has beenshown to be achieved with difference-intensity [24, 33, 37]single-mode intensity [33, 37] as well as balanced homo-dyne detection schemes [36, 37].The single-parameter QFI from equation (13) is foundto be F ( i ) = 4 | T | | α | (28)implying a QCRB∆ ϕ ( i ) QCRB = 12 | T || α | . (29)For the balanced case we get F ( i ) = 2 | α | ,thus we already improve the phases sensitivity,∆ ϕ ( i ) QCRB = 1 / √ | α | . We can even go further and con-sider the “unphysical” case | T | → ϕ ( i ) QCRB =1 / | α | . In Section VII A we will show that there is noth-ing unphysical about this scenario, it all depends on howwe intend to measure our phase sensitivity.In the symmetric phase shift case, equation (19) gives F ( ii ) = | α | (30)with the corresponding QCRB,∆ ϕ ( ii ) QCRB = 1 | α | . (31)In Fig. 4 we plot the three discussed QFIs versus | T | for | α | = 10. With F ( ii ) remaining constant, regardless ofthe value of T , the “true” phase sensitivity F (2 p ) peaksfor a balanced beam splitter yielding the well-known re-sult F (2 p ) = | α | [24, 28] while F ( i ) steadily grows reach-ing its maximum value F ( i ) max = 4 | α | for | T | = 1. B. Double coherent input
In this scenario we consider the input state | ψ in i = | α β i = ˆ D ( α ) ˆ D ( β ) | i (32) FIG. 4. The three QFIs versus the transmission coefficient of BS for a single coherent input state with | α | = 10. While F ( ii ) remains constant, irrespective of | T | , F ( i ) steadilygrows from 0 to 4 | α | . The two parameter QFI reaches itsoptimum in the balanced case (i. e. | T | = 0 . where we denote α = | α | e iθ α , β = | α | e iθ β and ∆ θ = θ α − θ β . The first Fisher matrix element yields F ss = | α | + | β | . From equation (A2) we have F dd = | α | + | β | (33)and the last Fisher matrix element yields F sd = (cid:0) | T | − | R | (cid:1) ( | β | − | α | ) − | T R || αβ | sin ∆ θ. (34)Using these results we get two-parameter QFI and its ex-pression is given in equation (E1). As proved in reference[34], for | α | , | β | and ∆ θ given, an optimum transmissioncoefficient exists and it is given by T (2 p ) opt = vuut
12 + sign( ̟ − ̟ sin ∆ θ q (1 − ̟ ) + 4 ̟ sin ∆ θ (35)where ̟ = | β | / | α | . Replacing | T | with T (2 p ) opt in equation(E1) (and assuming T (2 p ) opt = { , } ) brings F (2 p ) to itsglobal maximum, F (2 p ) max = | α | + | β | (36)implying the QCRB∆ ϕ (2 p ) QCRB = 1 p | α | + | β | . (37)In the asymmetric single phase shift case (see Fig. 2) thesingle-parameter QFI equation (13) yields F ( i ) = 4 | T | | α | + 4 | R | | β | + 8 | T R || αβ | sin ∆ θ (38)While for the two-parameter QFI, regardless of the valueof the input PMC (∆ θ ) we can find an optimum trans-mission coefficient (35) bringing us to the maximal QFI FIG. 5. The three QFIs versus the transmission coefficientof BS for a double coherent input state, with two phase-matching conditions. While for ∆ θ = 0, F ( i ) linearly growsfrom 4 | β | to 4 | α | , for ∆ θ = π/ | α | + | β | )for | T | = T ( i ) opt . Parameters used: | α | = 10, | β | = 5. (36), this is no longer true for F ( i ) . If sin ∆ θ = 0, then F ( i ) is maximized for | T | = 1 (if | α | > | β | ) or for T = 0 (if | α | < | β | ). If sin ∆ θ = 0, the optimal transmission coeffi-cient is given by equation (E2). For sin ∆ θ = 1 equation(E2) yields the simple expression T ( i ) opt = | α | p | α | + | β | . (39)Replacing this result into equation (38) takes F ( i ) to itsglobal maximum F ( i ) max = 4( | α | + | β | ) (40)and this implies the QCRB∆ ϕ ( i ) QCRB = 12 p | α | + | β | . (41)Finally, in the symmetrical case (see Fig. 3), from equa-tion (19) we get F ( ii ) = | α | + | β | (42)and it implies ∆ ϕ ( ii ) QCRB = 1 / p | α | + | β | . Remarkably,this QFI is totally immune to the input PMC and to thetransmission coefficient of BS .In Fig. 5 we plot the three QFI metrics against thetransmission coefficient of BS , | T | . We consider | α | > | β | and we first discuss the case ∆ θ = 0. While F ( ii ) remains constant irrespective of the values taken by T and ∆ θ , F ( i ) varies linearly from 4 | β | (for | T | = 0) to4 | α | (for | T | = 1). The two-parameter QFI attains itsmaximum value in the balanced case. In the extremecase | T | = 0 /
1, regardless of the value of ∆ θ , it reaches F (2 p ) = 4 | αβ | / ( | α | + | β | ) in agreement with equation(12). For ∆ θ = 0, the behavior of both F (2 p ) and F ( i ) changes.In the case of a two-parameter QFI, while the max-imum attainable QFI (36) remains unchanged, the op-timum transmission coefficient shifts from the balancedcase [34]. For ∆ θ = π/ T (2 p ) opt ≈ √ .
13. The asymmetric single-parameter QFI is maximized to F ( i ) max = 4( | α | + | β | ) forthe transmission coefficient T ( i ) opt ≈ √ . C. Coherent plus squeezed vacuum input
In this scenario we have the input state | ψ in i = | α ξ i = ˆ D ( α ) ˆ S ( ξ ) | i . (43)The squeezed vacuum is obtained by applying the unitaryoperator [13, 49]ˆ S m ( χ ) = e [ χ ∗ ˆ a m − χ (ˆ a † m ) ] / (44)to a mode m previously found in the vacuum state and wedenote χ = se iϑ . Usually s ∈ R + is called the squeezingfactor and ϑ denotes the phase of the squeezed state.For the input state from equation (43) we employed asqueezing with ξ = re iθ applied to the input port 0. Thefirst Fisher matrix element is F ss = | α | + sinh r/ F dd = (cid:0) | T | − | R | (cid:1) (cid:18) | α | + sinh r (cid:19) +4 | T R | (cid:0) sinh r + Υ + ( α, ξ ) (cid:1) (45)where the Υ + function is defined in equation (D1). Thelast Fisher matrix element yields F sd = (cid:0) | T | − | R | (cid:1) (cid:18) sinh r − | α | (cid:19) (46)and using equation (4) we get the two-parameter QFI, F (2 p ) = 4 | T R | (cid:0) sinh r + Υ + ( α, ξ ) (cid:1) +2 (cid:0) − | T R | (cid:1) sinh r | α | | α | + sinh r . (47)As discussed in reference [34], ifsinh r + Υ + ( α, ξ ) − r | α | | α | + sinh r > F (2 p ) is maximized in the balanced case and we get F (2 p ) = sinh r + Υ + ( α, ξ ) . (49)Imposing the optimum input PMC, namely,2 θ α − θ = 0 (50) FIG. 6. The three considered QFIs versus the transmissioncoefficient of BS for a coherent plus squeezed vacuum in-put state. As the squeezing factor r increases, the quantumadvantage of this state becomes obvious. Parameters used: | α | = 10 and the PMC 2 θ α − θ = 0. implies Υ + ( α, ξ ) = | α | e r and it maximizes the QFI tothe well-known result F (2 p ) max = sinh r + | α | e r [24, 28,37, 41].For the asymmetric single phase shift scenario fromfrom Fig. 2, the single-parameter QFI (13) becomes F ( i ) = 4 | T | | α | + 2 | R | sinh r +4 | T R | (cid:0) sinh r + Υ + ( α, ξ ) (cid:1) . (51) F ( i ) is maximized by an optimum transmission coefficient(see discussion in Appendix F) T ( i ) opt = s Υ + ( α, ξ ) − sinh r (1 + 2 cosh 2 r )2 (cid:0) Υ + ( α, ξ ) − | α | − sinh r cosh 2 r (cid:1) (52)The maximum single-parameter QFI, F ( i ) max , can then beobtained by replacing T ( i ) opt into equation (51) and theresult is given in equation (F4).In the symmetrical case from Fig. 3, equation (19)yields for the single parameter QFI F ( ii ) = (cid:0) − | T R | (cid:1) (cid:18) | α | + sinh r (cid:19) +2 | T R | (cid:0) sinh r + Υ + ( α, ξ ) (cid:1) . (53)If the conditionΥ + ( α, ξ ) − | α | − sinh r cosh 2 r > F ( ii ) is maximized in the balanced caseyielding F ( ii ) max = | α | + sinh r + sinh r + Υ + ( α, ξ )2 (55) FIG. 7. The three considered QFIs versus the transmissioncoefficient of BS for a squeezed-coherent plus squeezed va-cuum input state. The enhancement brought by the secondsqueezer is obvious for F ( i ) , however insignificant for F (2 p ) .Parameters used: | α | = 10, r = 1 .
2, 2 θ α − θ = 0, 2 θ α − φ = π . The three considered QFIs are plotted in Fig. 6 versusthe transmission coefficient of BS for two squeezing fac-tors. One notes the poor performance of F ( ii ) , even withrespect to the two-parameter QFI. Both F (2 p ) and F ( ii ) yield their maximum value in the balanced scenario while F ( i ) peaks at T ( i ) opt given by equation (52).The “quantum advantage” becomes quite obvious ifwe compare Figs. 5 and 6. Although | β | > sinh r , thecoherent plus squeezed vacuum completely outperformsthe double coherent input in terms of maximum QFI.If the condition | α | ≫ sinh r is satisfied and the op-timum PMC (50) employed, equation (52) approximatesto T ( i ) opt ≈ e r / p e r −
1) (valid under the constraint T ( i ) opt ≤
1) implying the maximum single-parameter QFI, F ( i ) max ≈ e r e r − | α | . (56)For small squeezing factors there is an advantage in em-ploying the single parameter QFI (see Fig. 6, dashedlines), a fact also seen from the fact that F ( i ) max / F (2 p ) max ≈ e r / ( e r − ≈ . r = 0 . e r ≫
1, implying T ( i ) opt ≈ / √
2. We thus have F ( i ) max ≈ F (2 p ) max ≈ e r | α | . Inother words, there is only a marginal advantage of havingavailable an external phase reference in the high-intensityand strong squeezing regime for a coherent plus squeezedvacuum input. D. Squeezed-coherent plus squeezed vacuum input
In this scenario we have the input state | ψ in i = | ( αζ ) ξ i = ˆ D ( α ) ˆ S ( ζ ) ˆ S ( ξ ) | i (57) where we applied the squeezing operator (44) to port 1with the parameters ζ = ze iφ . The first Fisher matrixelement is found to be F ss = sinh r z − ( α, ζ ) (58)where the function Υ − is defined by equation (D1). Theother two Fisher matrix elements are detailed in Ap-pendix G. From these Fisher matrix elements we cancompute the three considered QFIs, results also givenin Appendix G.The two-parameter QFI (G3) reduces to the simpleexpression F (2 p ) = | α | e r + sinh ( r + z ) in the balancedcase with the optimal input PMCs [34, 36]: (cid:26) θ α − θ = 0 θ − φ = ± π. (59)Similar to the coherent plus squeezed vacuum case, wecan derive from equation (G4) an optimal transmissioncoefficient T ( i ) opt that maximizes F ( i ) (see Appendix G).The QFI describing the symmetrical ± ϕ/ BS trans-mission coefficient | T | . We considered two rather smallsqueezing factors ( z = 0 .
35 and, respectively, z = 0 . T ( i ) opt ≈ √ . z = 0 .
35) and T ( i ) opt ≈ √ .
83 (for z = 0 . F ( i ) around T ( i ) opt . These con-clusions also hold in the experimentally interesting high-intensity regime | α | ≫ { sinh r, sinh z } , where we canapproximate T ( i ) opt ≈ s | − e z − r ) | (60)and this expression is meaningful as a transmission factorwhile T ( i ) opt ≤ VI. REALISTIC DETECTION SCHEMES
We close now the Mach-Zehnder interferometer (MZI)with BS (characterized by the transmission/reflectioncoefficients T ′ /R ′ ) and discuss the performance of tworealistic detection schemes, namely the difference inten-sity and the balanced homodyne detection techniques(see Fig. 8). We consider the most general case (paral-leling the two-parameter Fisher estimation from SectionII) and assume two independent phase shifts ϕ (in thelower arm of the interferometer) and ϕ (in the upper FIG. 8. The two realistic detection schemes considered here:the difference-intensity detection with its associated operatorˆ N d and the balanced homodyne detection with its associatedoperator ˆ X L . one). Thus we can easily set ϕ = 0 and we have thescenario from Section III or we can set ϕ = − ϕ = ϕ/ A. Difference-intensity detection
A good example of a realistic detection scheme sen-sitive only to the difference phase shift ( ϕ − ϕ ) isthe difference-intensity detection scheme [33, 36, 37] (seeFig. 8). The observable conveying information about thephase shift is ˆ N d = ˆ a † ˆ a − ˆ a † ˆ a (61)and the final expression with respect to the input fieldoperators is given in equation (H3). The phase sensitivityis defined as usual,∆ ϕ df = p ∆ ˆ N d (cid:12)(cid:12)(cid:12) ∂ h ˆ N d ( ϕ ) i ∂ϕ (cid:12)(cid:12)(cid:12) (62)where ∂ h ˆ N d ( ϕ ) i /∂ϕ is given in equation (H4) and thevariance ∆ ˆ N d is given in equation (H5). B. Balanced homodyne detection
If we assume a balanced homodyne detection schemeat the output port 4 (see Fig. 8), the relevant operatormodeling this detection is given byˆ X φ L = e − iφ L ˆ a + e iφ L ˆ a † φ L (assumed fixed and controllable with respectto θ α ) is the phase of the local coherent source | γ i where γ = | γ | e iφ L . The final expression for ˆ X φ L with regard to the input field operators is given in Appendix I. We definethe phase sensitivity of a balanced homodyne detector as∆ ϕ hom = q ∆ ˆ X φ L (cid:12)(cid:12)(cid:12) ∂ h ˆ X φL i ∂ϕ (cid:12)(cid:12)(cid:12) (64)If we consider the scenario from Fig. 2, we have (cid:12)(cid:12)(cid:12)(cid:12) h ˆ X φ L i ∂ϕ (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) ℜ n e − i ( φ L + ϕ ) ( R h ˆ a i + T h ˆ a i ) o (cid:12)(cid:12)(cid:12) | R ′ | (65)and for the symmetric ± ϕ/ (cid:12)(cid:12)(cid:12)(cid:12) ∂ h ˆ X φ L i ∂ϕ (cid:12)(cid:12)(cid:12)(cid:12) = 12 ℜ n ie − iφ L (cid:16) T T ′ e iϕ/ − RR ′ e − iϕ/ (cid:17) h ˆ a i + ie − iφ L (cid:16) RT ′ e iϕ/ − T R ′ e − iϕ/ (cid:17) h ˆ a i o . (66)The final expression for the variance ∆ ˆ X φ L with respectto the input field operators is given in equation (I2). VII. PHASE SENSITIVITY COMPARISONWITH GAUSSIAN INPUT STATES
Paralleling the discussion from Section V, we comparehere the realistically achievable phase sensitivities for var-ious input Gaussian states versus the QCRBs implied bythe QFIs discussed before. We consider both detectionschemes presented in Section VI.
A. Single coherent input
From the phase sensitivity formula (62) and consider-ing the input state (23), for a difference-intensity detec-tion scheme we get∆ ϕ df = 14 | T RT ′ R ′ || α || sin ϕ | (67)and comparing this result with the QCRB from equation(26) we note that it can be attained only if BS is bal-anced. Moreover, this detection scheme yields the sameresult for the scenarios from Figs. 2 and 3. The phasesensitivity ∆ ϕ df from equation (67) is further optimizedif BS is balanced, too, yielding the well-known result[24, 33] ∆ ϕ df = 1 | α || sin ϕ | . (68)We note that for | T | → | T | →
0) the phase sensi-tivity degrades, a behavior expected from the vanishingof the two-parameter QFI from equation (12).For a balanced homodyne detection scheme we obtainthe variance ∆ ˆ X φ L = 1 /
4. In the setup from Fig. 2,from equation (65) we have (cid:12)(cid:12)(cid:12)(cid:12) h ˆ X φ L i ∂ϕ (cid:12)(cid:12)(cid:12)(cid:12) = | T R ′ || α || cos( φ L + ϕ − θ α ) | (69) FIG. 9. Phase sensitivity for a single coherent input. Thebalanced homodyne detection scheme approaches the QCRBscorresponding to F ( i ) and, respectively F ( ii ) . The perfor-mance of the difference intensity detection scheme is limitedby the QCRB corresponding to the two-parameter QFI. Pa-rameters used: | α | = 10 and φ L = θ α . and imposing φ L = θ α we end up with a phase sensitivity∆ ϕ ( i ) hom = 12 | T R ′ || α || cos ϕ | . (70)Contrary to ∆ ϕ df from equation (67), an unbalanced in-terferometer with | T | → | T ′ | → ϕ ( i ) QCRB from equation (15) if we select the optimalworking point where | cos ϕ | = 1. In the balanced case,the best phase sensitivity ∆ ϕ ( i ) hom = 1 / | α | is indeed limi-ted by ∆ ϕ (2 p ) QCRB and this explains why previous papers[36, 37] did not report sensitivities beyond the QCRBfrom equation (5).In the symmetrical scenario (see Fig. 3), from equation(66) we get (cid:12)(cid:12)(cid:12)(cid:12) ∂ h ˆ X φ L i ∂ϕ (cid:12)(cid:12)(cid:12)(cid:12) = | α | (cid:12)(cid:12) | T R ′ | cos ( φ L + ϕ/ − θ α ) −| T ′ R | cos( φ L − ϕ/ − θ α ) (cid:12)(cid:12) (71)and with the condition φ L − θ α = 0 we find the phasesensitivity∆ ϕ ( ii ) hom = 1 (cid:12)(cid:12) | T R ′ | − | RT ′ | (cid:12)(cid:12) | α | (cid:12)(cid:12) cos (cid:0) ϕ (cid:1) (cid:12)(cid:12) (72)It is obvious that max (cid:12)(cid:12) | T R ′ | − | RT ′ | (cid:12)(cid:12) ≤ | T | → | T ′ | → | T | → | T ′ | →
1. Assuming the first case we have | T R ′ | ≈ | T ′ R | ≈ ϕ opt = 2 kπ with k ∈ Z )∆ ˜ ϕ ( ii ) hom ≈ | α | (73) FIG. 10. Phase sensitivity for a dual coherent input. Thehomodyne detection approaches the QCRBs corresponding tothe single-parameter QFI. Parameters used: | α | = 10, | β | = 5.The difference-intensity detection scheme reaches the QCRBcorresponding to the two-parameter QFI. Please note the dif-ferent optimal working points for the considered detectionschemes. This phase sensitivity is indeed limited by the QCRB∆ ϕ ( ii ) QCRB from equation (20).In Fig. 9 we depict the performance of both detectionschemes versus the three QCRBs discussed before. Weplot the difference-intensity detection scheme at its bestperformance (implying both BS balanced). For the bal-anced homodyne detection scheme we consider the trans-mission coefficients T = 0 .
99 (for BS ) and T ′ = 0 .
01 (for BS ). As see from Fig. 9, all three QCRBs have a physi-cal meaning and with the appropriate setup can actuallybe attained. B. Double coherent input
With a double coherent input state (32) more degreesof freedom become available in order to outline the phys-ical meaning of each of the three QCRB.In the case of a difference-intensity detection schemewe have the phase sensitivity (see Appendix J)∆ ϕ df = p | α | + | β | (cid:12)(cid:12) | T R | sin ϕ ( β | − | α | ) + | αβ | cos ϕ (cid:12)(cid:12) | T ′ R ′ | (74)and we assumed here the optimal PMC ∆ θ = 0. Thephase sensitivity is further optimized for both BS bal-anced yielding its best performance at the working point∆ ˜ ϕ df = 1 p | α | + | β | (75)where the optimum working point ϕ opt is given byequation (J3) and we conclude that we can attain the0∆ ϕ (2 p ) QCRB from equation (37). We wish to point out thatin general, ϕ opt = kπ/ k ∈ Z .We emphasize now an interesting point about the dualcoherent input state, already mentioned in Section V B,namely that if we change the input PMC from ∆ θ = 0to ∆ θ = π/ F (2 p ) is decreased, however F ( i ) increases.While the decrease of F (2 p ) is less surprising and alreadydiscussed in the literature [33, 34], the increase of F ( i ) is somehow surprising and the attainability of its corre-sponding QCRB may raise some doubts.For a balanced homodyne detection scheme we get forthe variance ∆ ˆ X φ L = 1 /
4. In the case of an asymmetricphase shift (see Fig. 2), from equation (65) we get (cid:12)(cid:12)(cid:12)(cid:12) h ˆ X φ L i ∂ϕ (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12) | T | cos ϕ | α | + | R | sin(∆ θ + ϕ ) | β | (cid:12)(cid:12) | R ′ | (76)and we assumed φ L = θ α . If we further assume ∆ θ = π/ T ( i ) opt fromequation (39), we get the phase sensitivity∆ ϕ ( i ) hom = 12 | R ′ | p | α | + | β | | cos ϕ | (77)and for at the optimum working point ϕ opt = (2 k + 1) π ( k ∈ Z ) and | R ′ | → ϕ ( i ) hom → ∆ ϕ ( i ) QCRB .In the case of a symmetric phase shift (see Fig. 3),from equation (66) and imposing φ L = θ α , the optimaltransmission factor from equation (39) and ∆ θ = π/ ϕ ( ii ) hom = 1 | R ′ | p | α | + | β | (cid:12)(cid:12) cos ϕ (cid:12)(cid:12) (78)and we assumed | R ′ | ≫ | T ′ | . Imposing the optimumworking point ϕ opt = 2 kπ ( k ∈ Z ) and | R ′ | → ϕ ( ii ) hom → ∆ ϕ ( ii ) QCRB .In Fig. 10 we plot the three phase sensitivities againsttheir corresponding QCRBs. Similar to Section VII A,the difference-intensity detection scheme is consideredwith both BS balanced. For the balanced homodyne de-tection we consider T given by equation (39) and for thesecond BS we took T ′ = 0 .
01. One notes that each de-tection scheme approaches its corresponding QCRB.
C. Coherent plus squeezed vacuum input
In the following two sections we only consider the phasesensitivities ∆ ϕ ( i ) hom and ∆ ϕ (2 p ) hom . With the input stategiven by equation (43) we find (cid:12)(cid:12)(cid:12)(cid:12) ∂ h ˆ N d i ∂ϕ (cid:12)(cid:12)(cid:12)(cid:12) = 4 | T RT ′ R ′ || sin ϕ | (cid:12)(cid:12) | α | − sinh r (cid:12)(cid:12) (79)One notes that the balanced case (for both BS ) maxi-mizes this term. For the variance ∆ ˆ N d , we obtain theresult given in equation (K1). FIG. 11. Phase sensitivity for a coherent plus squeezed va-cuum input. Parameters used: | α | = 10, r = 1 . θ α − θ = 0. Inset: zoom around the peak sensitivity for thephase shift range ϕ/π ∈ [0 , ϕ df is largely subop-timal, ∆ ϕ ( i ) hom is much closer to optimality. In the case of a balanced homodyne detection, equa-tion (69) remains valid. The variance is given by equa-tion (K2) and combining these results takes us to thephase sensitivity from equation (K3). Further simplifica-tions are obtained by assuming φ L = θ α and the PMC(50) satisfied, yielding the phase sensitivity ∆ ϕ ( i ) hom fromequation (K4).Imposing the optimum working point ϕ opt = π takesus to the best achievable phase sensitivity,∆ ˜ ϕ ( i ) hom = p − ( | T T ′ | + | RR ′ | ) (1 − e − r )2 | T R ′ || α | . (80)We notice that we get the well-known result ∆ ˜ ϕ ( i ) hom = e − r / | α | [36, 37] by imposing both BS balanced. However,this is not the optimum setup. The best phase sensitivityis obtained by imposing the transmission coefficient T ( i ) opt from equation (52) to BS and T ′ ( i ) opt = T ( i ) opt r − (cid:16) T ( i ) opt (cid:17) (1 − e − r ) r − (cid:16) T ( i ) opt (cid:17) (1 − e − r ) (81)to BS .In Fig. 11 we depict the phase sensitivities as well asthe corresponding QCRBs versus the internal phase shift.For the difference-intensity detection scheme we consid-ered its optimal setup, i. e. with both BS balanced. Forthe homodyne detection scheme we considered the opti-mal transmission coefficients for the parameters used inFig. 11, namely T ( i ) opt ≈ √ .
53 and T ′ ( i ) opt ≈ √ . α regime is de-picted in Fig. 12. For small squeezing ( r = 0 . FIG. 12. Phase sensitivity for coherent plus squeezed vacuuminput in the high- α regime. Parameters used: | α | = 10 ,2 θ α − θ = 0 and θ − φ = π . case), ∆ ϕ ( i ) hom approaches ∆ ϕ ( i ) QCRB and shows notice-ably better performance than the ∆ ϕ (2 p ) QCRB . However,for a higher squeezing factor ( r = 1 . α regime with the con-straint of a low squeezing factor. D. Squeezed-coherent plus squeezed vacuum input
For the input state from equation (57) and a difference-intensity detection scheme we find (cid:12)(cid:12)(cid:12)(cid:12) ∂ h ˆ N d i ∂ϕ (cid:12)(cid:12)(cid:12)(cid:12) = 4 | T RT ′ R ′ | (cid:12)(cid:12) | α | + sinh z − sinh r (cid:12)(cid:12) | sin ϕ | (82)The variance ∆ ˆ N d can be obtained as before by applyingthe input state (57) to equation (H5). Throughout thissection we consider the input PMCs (59) satisfied. Thebalanced case for both BS and BS maximizes equation(82). The expression of ∆ ˆ N d for the balanced case canbe found in reference [36].For a balanced homodyne detection scheme, equation(69) remains valid. For φ L = θ α and PMCs (59) satisfied,the variance ∆ ˆ X L is given in equation (L1). Combiningthese findings and imposing the optimum working point ϕ opt = π takes us to the optimal phase sensitivity fromequation (L2).In Fig. 13 we plot the performance of both detectorsversus the internal phase shift. For the difference inten-sity detection scheme we considered both BS balancedwhile in the case of the balanced homodyne detection weapplied the optimal transmission factors T ( i ) opt ≈ √ . (cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9) FIG. 13. Phase sensitivity for a squeezed-coherent plussqueezed vacuum input versus the internal phase shift. Evenwith a modest squeezing factor in port 1, the advantage ofusing a squeezed-coherent plus squeezed vacuum input stateis obvious. Parameters used: | α | = 10, r = 1 . z = 0 .
75 andPMCs 2 θ α − θ = 0, θ − φ = π . Inset: zoom around the peaksensitivity for the phase shift range ϕ/π ∈ [0 , ϕ df is still suboptimal, ∆ ϕ ( i ) hom is very close to optimality. for BS and T ′ ( i ) opt ≈ √ .
17 for BS . We recall that T ( i ) opt stems from optimizing the QFI F ( i ) , while T ′ ( i ) opt was ob-tained by minimizing ∆ ϕ ( i ) hom and the result is given byequation (L3).As already noted in Section V C, for a coherent plussqueezed vacuum input, in the high- α regime with smallsqueezing, there is a certain advantage in using an exter-nal phase reference. However as the squeezing factor in-creases, we have F ( i ) max ≈ F (2 p ) max . This fact changed here,irrespective of the squeezing factor from port 0, thereis a sizable increase in the performance for a squeezed-coherent plus squeezed vacuum input state.In order to better outline this assertion, in Fig. 14 weplot on the same graphic the performance of coherentplus squeezed vacuum (i. e. z = 0) and squeezed-coherent plus squeezed vacuum inputs in the high- α regime.Thus, we conclude that having access to an externalphase reference for a squeezed-coherent plus squeezed va-cuum input state brings a gain in the phase sensitivity,gain that does not fade away with the increase of thecoherent amplitude, | α | . VIII. DISCUSSION
For a single coherent input state and a balanced ho-modyne detection scheme we obtained in Section VII Athe “unphysical” limits | T | → | T ′ | → ϕ ( i ) QCRB = 1 / | α | implied by F ( i ) .However, by analyzing Fig. 8, there is absolutely nothing2 FIG. 14. Phase sensitivity comparison between coherent plussqueezed vacuum and squeezed-coherent plus squeezed va-cuum input in the high- α regime. Parameters used: | α | = 10 , r = 1 .
2, 2 θ α − θ = 0 and θ − φ = π . unphysical about these limits. Indeed, we can write theinput state (including the local oscillator) as | Ψ i = | ψ in i ⊗ | γ i = | e iθ α | α |i ⊗ | e iφ L | γ |i (83)and we have an interferometer with one arm compris-ing the input port 1, through BS (total transmission),phase shift ϕ , BS (total reflection) and to BS L whilethe other arm is simply the local oscillator fed into thehomodyne’s balanced beam splitter. Since the two in-put signals have a fixed phase relation, interference isto be expected. In the case of the dual coherent inputfrom Section VII B, BS is no more in total transmis-sion/reflection mode, its transmission coefficient T ( i ) opt be-ing given by equation (39). When squeezing is added inone or both inputs, ∆ ϕ (2 p ) QCRB can be outperformed and∆ ϕ ( i ) QCRB approached with both BS and BS havingwell defined values of their respective transmission coef-ficients ( | T | , | T ′ | 6 = { , } ), as discussed in the previoussections.In reference [30] it was claimed that: “First, if botharms of the MZI have different unknown phase shifts inthe application and the input to one of the two ports isvacuum, then no matter what the input in the other portis, and no matter the detection scheme, one can neverbetter the SNL in phase sensitivity. [. . . ] This type ofsensing includes gravitational wave detection [. . . ] ” . In-deed, treating this as a two-parameter problem and im-posing the vacuum state for input port 0, equation (4)yields F (2 p ) = 4 | T R | h ˆ n i and since 4 | T R | ≤ ϕ = ϕ/ ϕ = − ϕ/ F ( ii ) applies, not F (2 p ) . With theinput port 0 in the vacuum state, equation (C1) yields F ( ii ) = ( | T | + | R | )∆ ˆ n + 2 | T R | h ˆ n i (84) and we have a sub-SNL sensitivity if ∆ ˆ n > h ˆ n i . Wewould also like to point out that gravitational waves withthe + polarization along any of the arms of the detectorand arriving perpendicular to the plane of the interfero-meter yield highly (anti)correlated phase shifts [52, 53].It has been argued that the external phase reference(the homodyne in our case) must be strong compared tothe other sources (e. g. | γ | ≫ {| α | , sinh r } for a coher-ent plus squeezed vacuum input). Thus, one might objectthat this scheme is irrelevant since it requires even moreresources. There are two arguments against this objec-tion. Sometimes the sample that causes the phase shiftinside the interferometer is delicate, as in the case of amicroscope [38, 39]. Thus, the available power shone onthe sample has to be drastically limited and any phasesensitivity enhancement via an external phase referenceis more than welcome. Second, when the interferometeris at its optimum working point, the average photon num-ber at output port 4 (for the single-mode intensity andbalanced homodyne detection schemes) is low for manyinput states [33]. Thus, the external phase reference canactually have a much lower amplitude than initially an-ticipated.Although losses are outside the scope of this paper,we can schematically discuss the effects of non-ideal pho-ton detectors [36, 54–56] and/or internal losses [56–58].Non-ideal photo-detection can be modeled by insertinga ficticious BS with a transmission factor η ( η = 1 im-plies no losses) in front of a ideal photo-detector [36, 54–56]. In the case of coherent states, we have the scaling∆ ϕ lossy = 1 / √ η ∆ ϕ ideal [36, 54, 55]. Thus, for mod-ern, high-efficient photo-detectors the effect should bemarginal. The impact is more severe in the case of a co-herent plus squeezed vacuum input [36, 55, 56] and in thecase of high losses the scaling approaches the SNL. For asqueezed-coherent plus squeezed vacuum input a similarpattern emerges [36]. However, workarounds have beenshown to exist. Wu, Toda & Hofmann [59] showed thatby using photon-number-resolving detectors (PNRDs) inthe dark port of an interferometer fed by a coherent plussqueezed vacuum input, up to a certain level of losses,the quantum Cram´er-Rao bound can be attained. In thecase of coherent light input, internal losses have the sameeffect as non-ideal photodetectors, while for a coherentplus squeezed vacuum input state they impact more theQFI terms that could have lead to a Heisenberg scaling[56]. IX. CONCLUSIONS
In this paper we reconsidered the single-parameter QFIversus the two-parameter one for an unbalanced MZI. Wetheoretically calculated the single parameter QFI bothfor a asymmetric and symmetric phase shifts scenarios aswell as the two-parameter QFI. From these QFIs we caninfer their corresponding quantum Cram´er-Rao bounds,implying the best achievable phase sensitivities.3Using a balanced homodyne detection technique andvarious Gaussian input states, we show that far from be-ing unphysical, the QCRB implied by the single parame-ter QFI is actually meaningful for each and every consid-ered input state. We find that a coherent plus squeezedvacuum input state can benefit from the availability of anexternal phase reference for a low squeezing factor and ahigh coherent amplitude if a properly unbalanced inter-ferometer is used. The restriction on the squeezing fac-tor(s) disappears for a squeezed-coherent plus squeezedvacuum input, this state being probably the most inter-esting candidate to demonstrate the sizable enhancementthat can be obtained by using an unbalanced interfero-meter and an external phase reference.We conclude that when assessing the “resources thatare actually not available” one must carefully ponder theactual experimental setup. If an external phase referenceis possible (through e. g. homodyne detection), then thesingle parameter quantum Fisher information might givethe pertinent answer regarding the best possible phase sensitivity.
ACKNOWLEDGMENTS
This work has been supported by the Extreme LightInfrastructure Nuclear Physics (ELI-NP) Phase II, aproject co-financed by the Romanian Government andthe European Union through the European Regional De-velopment Fund and the Competitiveness OperationalProgramme (1/07.07.2016, COP, ID 1334).
Appendix A: Two parameter Fisher information
Using the field operator transformations (6) we findthe (photon) number operator ˆ n = ˆ a † ˆ a , namelyˆ n = | R | ˆ a † ˆ a + | T | ˆ a † ˆ a − T ∗ R (cid:16) ˆ a † ˆ a − ˆ a ˆ a † (cid:17) (A1)and similarly ˆ n can be deduced. Starting from equation(2) and using the field operator transformations (6), aftersome calculations we arrive at the expression: F dd = (cid:0) | T | − | R | (cid:1) (cid:0) ∆ ˆ n + ∆ ˆ n (cid:1) + 8 | T R | (cid:16) h ˆ n ih ˆ n i − |h ˆ a i| |h ˆ a i| − ℜ n h (ˆ a † ) ih ˆ a i − h ˆ a † i h ˆ a i o(cid:17) +4 | T R | ( h ˆ n i + h ˆ n i ) − | T R | (cid:0) | T | − | R | (cid:1) (cid:16) ℑ n(cid:16) h ˆ a † ˆ n i − h ˆ a † ih ˆ n i (cid:17) h ˆ a i + h ˆ a i (cid:16) h ˆ a † ˆ n i − h ˆ n ih ˆ a † i (cid:17)o(cid:17) . (A2)In the balanced case F dd simplifies and the result is given by equation (8). The Fisher matrix term F sd is found to be F sd = (cid:0) | T | − | R | (cid:1) (cid:0) ∆ ˆ n − ∆ ˆ n (cid:1) + 4 | T R |ℑ n h ˆ a ih ˆ a † i + ( h ˆ n ˆ a i − h ˆ n ih ˆ a i ) h ˆ a † i + h ˆ a i (cid:16) h ˆ a † ˆ n i − h ˆ a † ih ˆ n i (cid:17)o . (A3) Appendix B: Single parameter Fisher information F ( i ) Starting from definition (13) and using the field operator transformations (6), after a number of calculations wearrive at the final result: F ( i ) = 4 | R | ∆ ˆ n + 4 | T | ∆ ˆ n + 4 | T R | (cid:0) h ˆ n i + h ˆ n i + 2( h ˆ n ih ˆ n i − |h ˆ a i| |h ˆ a i| ) (cid:1) − | T R | ℜ n h ˆ a ih (ˆ a † ) i − h ˆ a ih ˆ a † i o − | T R |ℑ n h ˆ a ih ˆ a † i o − | T R || R | ℑ n ( h ˆ n ˆ a i − h ˆ n ih ˆ a i ) h ˆ a † i o − | T R || T | ℑ n h ˆ a i (cid:16) h ˆ a † ˆ n i − h ˆ n ih ˆ a † i (cid:17)o . (B1)In the balanced case, F ( i ) reduces to F ( i ) = ∆ ˆ n + ∆ ˆ n + h ˆ n i + h ˆ n i + 2( h ˆ n ih ˆ n i − |h ˆ a i| |h ˆ a i| ) − ℜ n h ˆ a ih (ˆ a † ) i − h ˆ a ih ˆ a † i o − ℑ n h ˆ a ih ˆ a † i + ( h ˆ n ˆ a i − h ˆ n ih ˆ a i ) h ˆ a † i + h ˆ a i (cid:16) h ˆ a † ˆ n i − h ˆ n ih ˆ a † i (cid:17)o . (B2)4 Appendix C: Single parameter Fisher information F ( ii ) From definition (19), using the field operator transformations (6) we arrive at F ( ii ) = ( | T | + | R | )(∆ ˆ n + ∆ ˆ n ) + 2 | T R | (cid:0) h ˆ n i + h ˆ n i + 2( h ˆ n ih ˆ n i − |h ˆ a i| |h ˆ a i| ) (cid:1) − | T R | (cid:16) h ˆ a ih (ˆ a † ) i + h (ˆ a † ) ih ˆ a i − h ˆ a i h ˆ a † i − h ˆ a † i h ˆ a i (cid:17) +2 T ∗ R ( | T | − | R | ) (cid:16) h ˆ a † ˆ n i − h ˆ a † ih ˆ n i (cid:17) h ˆ a i − T ∗ R ( | T | − | R | ) ( h ˆ n ˆ a i − h ˆ n ih ˆ a i ) h ˆ a † i +2 T ∗ R ( | T | − | R | ) h ˆ a i (cid:16) h ˆ a † ˆ n i − h ˆ a † ih ˆ n i (cid:17) − T ∗ R ( | T | − | R | ) h ˆ a † i ( h ˆ n ˆ a i − h ˆ n ih ˆ a i ) . (C1)In the balanced case F ( ii ) simplifies to F ( ii ) = 12 (cid:16) ∆ ˆ n + ∆ ˆ n + h ˆ n i + h ˆ n i + 2( h ˆ n ih ˆ n i − |h ˆ a i| |h ˆ a i| ) − ℜ n h ˆ a ih (ˆ a † ) i − h ˆ a i h ˆ a † i o(cid:17) . (C2) Appendix D: The Υ functions We define the functions Υ + / − ( γ, χ ) = | γ | (cosh 2 s ± sinh 2 s cos (2 θ γ − ϑ )) (D1)with both arguments complex, γ = | γ | e iθ γ and χ = se iϑ with s ∈ R + , θ γ , ϑ ∈ [0 , π ]. These functions allow thecompact writing of Fisher matrix coefficients as well as output variances for a range of Gaussian input states [36].For the PMC 2 θ γ − ϑ = 0 we find Υ + ( γ, χ ) = | γ | e s and Υ − ( γ, χ ) = | γ | e − s . For the PMC 2 θ γ − ϑ = ± π we findΥ + ( γ, χ ) = | γ | e − s and Υ − ( γ, χ ) = | γ | e s . See also Fig. 2 in reference [36]. Appendix E: QFI calculations for a double coherent input
From equation (4) and using the results from equations (33) and (34) we get F (2 p ) = 4 | T R | (cid:0) | α | + | β | (cid:1) − | T R | | αβ | sin ∆ θ | α | + | β | + 4 (cid:0) | T | − | R | (cid:1) | αβ | | α | + | β | − | T R | (cid:0) | T | − | R | (cid:1) | αβ | | α | − | β | | α | + | β | sin ∆ θ. (E1)In the asymmetric single phase scenario from Fig. 2, an optimum transmission coefficient for BS (in the sense thatit maximizes F ( i ) ) can be found, T ( i ) opt = 1 r ( | α | −| β | ) | αβ | sin ∆ θ − || α | −| β | || αβ || sin ∆ θ | q ( | α | −| β | ) | αβ | sin ∆ θ . (E2) Appendix F: QFI calculations for a coherent plus squeezed vacuum input
The QFI F ( i ) from both equations (51) and (G4) can be put in the form F i = A f | T | + B f | R | + C f | T R | , i. e. F i = ( A f + B f − C f ) T + ( C f − B f ) T + B f (F1)and without loss of generality, starting from equation (F1) we assume T real. Differentiating with respect to T andsolving this equation brings us to T ( i ) opt = s C f − B f C f − A f + B f ) . (F2)5For the input state (43) we have the coefficients A f = | α | B f = sinh r C f = sinh r + Υ + ( α, ξ ) (F3)and we arrive at the expression given by equation (52). If T ( i ) opt exists, replacing (52) into equation (51) yields themaximum single-parameter QFI F ( i ) max = (Υ + ( α, ξ )) + sinh r Υ + ( α, ξ ) − r | α | + sinh r Υ + ( α, ξ ) − | α | − sinh r cosh 2 r . (F4)When discussing the conditions of existence of 0 ≤ T ( i ) opt ≤ r → F ( i ) = T | α | ). In the following we assume PMC (50) satisfied.We first define the limits: ( α lim = (cosh 2 r − r +0 . e r α lim = sinh r | − e r | (F5)For small values of r we have α lim < α lim . Thus, T ( i ) opt exists if | α | ∈ [ α lim , α lim ]. If α lim < α lim and moreover r ≤ ln 2 / T ( i ) opt exists if | α | ∈ [ α lim , α lim ]. Finally if r > ln 2 /
2, then T ( i ) opt exists for | α | ≥ α lim . Appendix G: QFI calculations for a squeezed-coherent plus squeezed vacuum input
For a squeezed-coherent state at input port 1 we have ∆ h ˆ n i = sinh z + Υ − ( α, ζ ) [36] and we employed this resultin computing F ss from equation (58). Using the input state (57) and the definition of the Fisher matrix element F dd we get F dd = (cid:0) | T | − | R | (cid:1) (cid:18) sinh r z − ( α, ζ ) (cid:19) + 4 | T R | (cid:0) Υ + ( α, ξ ) + sinh r + sinh z +2 sinh r sinh z (sinh r sinh z − cosh r cosh z cos( φ − θ )) (cid:1) (G1)Finally, starting from equation (A3), F sd is found to be F sd = (cid:0) | T | − | R | (cid:1) (cid:18) sinh r − sinh z − Υ − ( α, ζ ) (cid:19) (G2)From definition (4) and the previous results, we get the two-parameter QFI, F (2 p ) = 4 | T R | (cid:18) Υ + ( α, ξ ) + cosh 2 r cosh 2 z − sinh 2 r sinh 2 z cos( φ − θ ) − (cid:19) + (cid:0) | T | − | R | (cid:1) sinh r (cid:0) sinh z + 2Υ − ( α, ζ ) (cid:1) sinh r + sinh z + Υ − ( α, ζ ) . (G3)For the asymmetric phase shift case form Fig. 2, the single-parameter QFI yields F ( i ) = 4 | T | (cid:18) sinh z − ( α, ζ ) (cid:19) + 4 | R | sinh r | T R | (cid:0) Υ + ( α, ξ ) + sinh r + sinh z + 2 sinh r sinh z (sinh r sinh z − cosh r cosh z cos( φ − θ )) (cid:1) . (G4)If an optimal transmission factor 0 < T ( i ) opt < F ( i ) ), then it is given by equation(F2) with the coefficients A f = sinh z + Υ − ( α, ζ ) B f = sinh r C f = Υ + ( α, ξ ) + sinh r + sinh z + 2 sinh r sinh z (sinh r sinh z − cosh r cosh z cos( φ − θ )) (G5)6In Section V C we concluded that the Υ + ( α, ξ ) term dominates all other terms from equations (47) and (51) in thehigh- α regime. This assertion is still true for F (2 p ) from equation (G3). But, as mentioned in Section V D, in thehigh- α regime F ( i ) does not necessarily maximize in the balanced case. Indeed, F ( i ) from equation (G4) approximatesin this regime to F ( i ) ∼ | T | Υ − ( α, ζ ) + 4 | T R | Υ + ( α, ξ ) . (G6)and we immediately find the optimum transmission coefficient, T ( i ) opt ≈ s Υ + ( α, ξ )2 | Υ + ( α, ξ ) − Υ − ( α, ζ ) | (G7)For the optimum PMCs (59) satisfied, we arrive at T ( i ) opt from equation (60). If, from equation (60) one obtains T ( i ) opt >
1, this simply implies that the optimum transmission factor for BS is T ( i ) opt = 1.For the symmetric phase shift scenario we have the QFI F ( ii ) = sinh r z − ( α, ζ ) + 2 | T R | (cid:18) Υ + ( α, ξ ) − Υ − ( α, ζ ) − sinh r cosh 2 r − sinh z cosh 2 z + 2 sinh r sinh z (sinh r sinh z − cosh r cosh z cos( φ − θ )) (cid:19) (G8)and if the conditionΥ + ( α, ξ ) − Υ − ( α, ζ ) − sinh r cosh 2 r − sinh z cosh 2 z + 2 sinh r sinh z (sinh r sinh z − cosh r cosh z cos( φ − θ )) > F ( ii ) maximizes in the balanced case. Appendix H: Difference-intensity detection
From Fig. 8, using the field operator transformations (6) and (cid:26) ˆ a = R ′ ˆ a ′ + T ′ ˆ a ′ ˆ a = T ′ ˆ a ′ + R ′ ˆ a ′ (H1)where we recall that T ′ ( R ′ ) denote the transmission (reflection) coefficients of BS , we can write the field operatortransformations ˆ a = e − iϕ (cid:2)(cid:0) T T ′ + RR ′ e − iϕ (cid:1) ˆ a + (cid:0) T R ′ e − iϕ + RT ′ (cid:1) ˆ a (cid:3) ˆ a = e − iϕ (cid:2)(cid:0) T R ′ + RT ′ e − iϕ (cid:1) ˆ a + (cid:0) T T ′ e − iϕ + RR ′ (cid:1) ˆ a (cid:3) (H2)where ϕ = ϕ − ϕ . The final expression for ˆ N d isˆ N d = (cid:0) ( | T | − | R | )( | T ′ | − | R ′ | ) − | T RT ′ R ′ | cos ϕ (cid:1) (ˆ n − ˆ n )+2 (cid:0) T ∗ R ( | R ′ | − | T ′ | ) + ( | R | e − iϕ − | T | e iϕ ) T ′∗ R ′ (cid:1) ˆ a ˆ a † +2 (cid:0) T ∗ R ( | T ′ | − | R ′ | ) + ( | T | e − iϕ − | R | e iϕ ) T ′∗ R ′ (cid:1) ˆ a † ˆ a (H3)One can see from equation (H3) that ˆ N d depends only on ϕ = ϕ − ϕ thus insensitive to an external (or global)phase. The derivative of h ˆ N d i with respect to ϕ yields ∂ h ˆ N d i ∂ϕ = 4 (cid:16) | T R | sin ϕ ( h ˆ n i − h ˆ n i ) + ℜ n ( | R | e − iϕ + | T | e iϕ ) h ˆ a ih ˆ a † i o(cid:17) | T ′ R ′ | (H4)For the variance we find ∆ ˆ N d = A d (∆ ˆ n + ∆ ˆ n ) + | C d | ( h ˆ n i + h ˆ n i ) + 2 | C d | ( h ˆ n ih ˆ n i − |h ˆ a i| |h ˆ a i| )+2 ℜ n C d ( h ˆ a ih (ˆ a † ) i − h ˆ a i h ˆ a † i ) o + 4 A d ℜ n C d (cid:16) ( h ˆ n ˆ a i − h ˆ n ih ˆ a i ) h ˆ a † i − h ˆ a i ( h ˆ a † ˆ n i − h ˆ n ih ˆ a † i ) (cid:17)o (H5)7where we made the notations (cid:26) A d = 1 − | T || R ′ | + | R || T ′ | ) + 4 | T R || T ′ R ′ | (1 − cos ϕ ) C d = 2 | T ′ R ′ | sin ϕ + 2 i (cid:0) | T R | ( | R ′ | − | T ′ | ) + (1 − | T | ) | T ′ R ′ | cos ϕ (cid:1) (H6)and by direct calculation we also find the constraint A d + | C d | = 1 . (H7) Appendix I: Balanced homodyne detection
From equations (H2) and using the definition of ˆ X φ L we have h ˆ X φ L i = ℜ (cid:8) e − iφ L (cid:0)(cid:0) T T ′ e − iϕ + RR ′ e − iϕ (cid:1) h ˆ a i + (cid:0) T R ′ e − iϕ + RT ′ e − iϕ (cid:1) h ˆ a i (cid:1)(cid:9) (I1)For the asymmetric phase shift scenario from Fig. 2 we have ϕ = ϕ and ϕ = 0. The derivative of h ˆ X φ L i withrespect to ϕ gives the expression from equation (65). For the symmetric scenario from Fig. 3 we have ϕ = ϕ/ ϕ = − ϕ/
2, thus we get the result from equation (66). The variance of ˆ X φ L is found to be∆ ˆ X φ L = 14 + 2 ℜ (cid:8) A ∆ ˆ a + B ∆ ˆ a (cid:9) + 2 | A | ( h ˆ n i − |h ˆ a i| ) + 2 | B | ( h ˆ n i − |h ˆ a i| ) (I2)where we have the coefficients (cid:26) A = e − i ( φ L + ϕ ) (cid:0) T T ′ + RR ′ e − iϕ (cid:1) B = e − i ( φ L + ϕ ) (cid:0) T R ′ e − iϕ + T ′ R (cid:1) . (I3) Appendix J: Phase sensitivity calculations for a dual coherent input
For difference intensity detection scheme and a dual coherent input we get (cid:12)(cid:12)(cid:12)(cid:12) ∂ h ˆ N d i ∂ϕ (cid:12)(cid:12)(cid:12)(cid:12) = 4 (cid:12)(cid:12) | T R | sin ϕ ( β | − | α | ) + | αβ | (cid:0) | R | cos( θ α − θ β + ϕ ) + | T | cos( θ α − θ β − ϕ (cid:1) (cid:12)(cid:12) | T ′ R ′ | (J1)and the variance is simply ∆ ˆ N d = | α | + | β | . The phase sensitivity is optimized for the input PMC ∆ θ = 0 andequation (J1) becomes (cid:12)(cid:12)(cid:12)(cid:12) ∂ h ˆ N d i ∂ϕ (cid:12)(cid:12)(cid:12)(cid:12) = 4 (cid:12)(cid:12) | T R | sin ϕ ( β | − | α | ) + | αβ | cos ϕ (cid:12)(cid:12) | T ′ R ′ | . (J2)For T given, this expression is maximized at the optimum internal phase sift ϕ opt = arccos | αβ | p | αβ | + | T R | ( β | − | α | ) ! (J3)yielding the phase sensitivity at the optimum angle∆ ˜ ϕ df = p | α | + β | p | αβ | (1 − | T R | ) + | T R | ( | α | + β | ) | T ′ R ′ | . (J4)We can further optimize ∆ ˜ ϕ df by imposing both BS and BS balanced and we arrive at the expression (75). Appendix K: Phase sensitivity calculations for a coherent plus squeezed vacuum input
For the variance of the output number operator we find the expression∆ ˆ N d = A d sinh r | α | + | C d | (sinh r + 2 | α | sinh r ) − sinh 2 r | α | ℜ n C d e − i (2 θ α − θ ) i o (K1)8where the coefficients A d and C d were defined in equation (H6). The expression of ∆ ˆ N d for the balanced case canbe found in the literature [24, 33].In case of a balanced homodyne detection, the variance ∆ ˆ X φ L from equation (I2) becomes∆ ˆ X φ L = 14 + 2 ℜ (cid:8) A ∆ ˆ a (cid:9) + 2 | A | h ˆ n i (K2)and the phase sensitivity is found to be∆ ϕ ( i ) hom = q + 2 | A | sinh r − ℜ { A e iθ } sinh 2 r | T R ′ || α || cos ϕ | . (K3)Assuming φ L = θ α and the PMCs from equation (50) satisfied, the phase sensitivity can be written as∆ ϕ ( i ) hom = p − | T T ′ | + | RR ′ | ) sinh re − r + | RR ′ | sinh 2 r (1 − cos(2 ϕ )) + 4 | T RT ′ R ′ | sinh re − r (1 + cos ϕ )2 | T R ′ || α || cos ϕ | . (K4) Appendix L: Phase sensitivity calculations for a squeezed-coherent plus squeezed vacuum input
Assuming φ L = θ α and the PMCs (59) satisfied, the variance of the operator ˆ X L is found to be∆ ˆ X φ L = 14 −
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