Optimal Distributed quantum sensing using Gaussian states
OOptimal distributed quantum sensing using Gaussian states
Changhun Oh, Changhyoup Lee, Seok Hyung Lie, and Hyunseok Jeong Department of Physics and Astronomy, Seoul National University, Seoul 08826, Korea Institute of Theoretical Solid State Physics, Karlsruhe Institute of Technology, Karlsruhe 76131, Germany (Dated: March 10, 2020)We find and investigate the optimal scheme of distributed quantum sensing using Gaussian statesfor estimation of the average of independent phase shifts. We show that the ultimate sensitivity isachievable by using an entangled symmetric Gaussian state, which can be generated using a single-mode squeezed vacuum state, a beam-splitter network, and homodyne detection on each outputmode in the absence of photon loss. Interestingly, the maximal entanglement of a symmetric Gaus-sian state is not optimal although the presence of entanglement is advantageous as compared tothe case using a product symmetric Gaussian state. It is also demonstrated that when loss occurs,homodyne detection and other types of Gaussian measurements compete for better sensitivity, de-pending on the amount of loss and properties of a probe state. None of them provide the ultimatesensitivity, indicating that non-Gaussian measurements are required for optimality in lossy cases.Our general results obtained through a full-analytical investigation will offer important perspectivesto the future theoretical and experimental study for distributed Gaussian quantum sensing.
I. INTRODUCTION
Quantum resources are known to be useful for fur-ther enhancing the precision and the sensitivity of es-timation of various physical quantities beyond the stan-dard quantum limit [1–5]. A number of studies on single-parameter estimation have been performed over the lastfew decades [6], but much attention has begun to bepaid to estimation of multiparameters in recent years [7].Quantum-enhanced sensitivity in simultaneous estima-tion of multiple phases has been investigated to explainthe role of quantum entanglement and identify optimaland realistic setups saturating the ultimate theoreticalsensitivity [8–12]. The advantage of exploiting quan-tum entanglement becomes more significant when sens-ing takes place in different locations and the parameter ofinterest is a global feature of the network, e.g., the aver-age of distributed independent phases [13–18]. Such dis-tributed sensing is related to applications such as globalclock synchronization [19] and phase imaging [8]. Theseinspire the use of more practical quantum resources thatare feasible in a well-controlled manner with current tech-nology, e.g., Gaussian systems [20]. Very recently, thesensitivities of distributed quantum sensing with Gaus-sian states were studied under specific conditions [16, 17].The ultimate sensitivity and feasible optimal schemes,however, are not yet found and studied in the class ofGaussian metrology [21–23].In this paper, we investigate the ultimate sensitivityfor the average phase estimation in distributed quantumsensing with Gaussian states, where the phases are en-coded onto a multimode Gaussian probe state, as de-scribed in Fig. 1. We find an optimal probe state andmeasurement setup that achieve the ultimate sensitivity,which are shown to be experimentally feasible with cur-rent technology. Interestingly, we demonstrate that theoptimal symmetric Gaussian probe state is not a max-imally entangled state. For practical relevance, we fur-ther analyze the effect of loss, the entanglement-enhanced gain, and other Gaussian measurements in various con-ditions.We begin with a brief introduction to the formalismdescribing Gaussian states and multiparameter estima-tion. Gaussian states are defined as states whose Wignerfunctions are Gaussian distributions, and thus charac-terized by the first moment vector d i = Tr[ˆ ρ ˆ Q i ] andthe covariance matrix Γ ij = Tr[ˆ ρ { ˆ Q i − d i , ˆ Q j − d j } / { ˆ A, ˆ B } ≡ ˆ A ˆ B + ˆ B ˆ A . Here, a quadrature operatorvector of a M -mode continuous variable quantum systemis defined as ˆ Q = (ˆ x , ˆ p , ..., ˆ x M , ˆ p M ) T , satisfying thecanonical commutation relation, [ ˆ Q j , ˆ Q k ] = i ( Ω M ) jk ,where Ω M = (cid:16) − (cid:17) ⊗ M and M is the M × M iden-tity matrix. II. DISTRIBUTED SENSING
Consider estimation of M -parameter φ =( φ , φ , ..., φ M ) T based on measurement outcomes x , ob-tained with a conditional probability p ( x | φ ). The multi-parameter Cram´er-Rao inequality states that the M × M estimation error matrix Σ ij = (cid:104) ( ˆ φ i − φ i )( ˆ φ j − φ j ) (cid:105) ofany unbiased estimator ˆ φ i is bounded by the Fisherinformation matrix (FIM), F ( φ ), i.e., Σ ≥ F − , where F ij ( φ ) = (cid:80) x p ( x | φ ) ∂p ( x | φ ) ∂φ i ∂p ( x | φ ) ∂φ j [24]. Theconditional probability p ( x | φ ) = Tr[ˆ ρ φ ˆΠ x ] is givenby a positive operator-valued measure ˆΠ x for a givenparameter-encoded state ˆ ρ φ . The quantum Cram´er-Rao inequality sets a lower bound for the errorof an unbiased estimator, i.e., Σ ≥ F − ≥ H − ,where H ij = Tr[ˆ ρ φ { ˆ L i , ˆ L j } ] / L i being a sym-metric logarithmic derivative operator associatedwith i th parameter φ i [25]. When a linear combinationof φ i ’s, φ ∗ = w T φ = (cid:80) Mi =1 w i φ i with the weight a r X i v : . [ qu a n t - ph ] M a r FIG. 1. Schematic of distributed sensing under investigation.A multimode probe state ˆ ρ probe generated from the first beamsplitter network (BSN) for a given product state input ⊗ Mi =1 ˆ ρ i undergoes the individual phase shifts on each mode. Theparameter-imprinted state ˆ ρ φ is fed into the second BSN, fol-lowed by measurement. The measurement outcomes are usedin post-processing to estimate the parameter φ ∗ = (cid:80) Mi =1 w i φ i with the weight vector w . vector w , is of particular interest, the estimation erroris bounded as [26]∆ φ ∗ ≡ (cid:104) ( ˆ φ ∗ − φ ∗ ) (cid:105) ≥ w T F − w ≥ w T H − w . (1)Here, F − and H − are understood as the inverse ontheir support if the matrices are singular. Throughoutthis paper, we assume the normalization (cid:80) Mi =1 | w i | = 1for simplicity. III. GAUSSIAN DISTRIBUTED SENSORA. Quantum Fisher information matrix
Consider a distributed phase sensor in which a prod-uct Gaussian input state ⊗ Mi =1 ˆ ρ i is injected into a beamsplitter network (BSN), preparing a probe state ˆ ρ probe ,the multiphase information is encoded onto ˆ ρ probe by aunitary operation ˆ U φ = exp( − i (cid:80) Mj =1 φ j ˆ a † j ˆ a j ), and theoutput state ˆ ρ φ is measured after the second BSN, asdepicted in Fig. 1. Note that configuration of the firstBSN enables one to generate any probe Gaussian states[22, 27]. We also implicitly assume a strong referencebeam to define the phases, accessible in each mode formeasurement [28]. Here, we aim to investigate the sen-sitivity of Gaussian states for estimation of the parame-ter φ ∗ . When the probe state ˆ ρ probe after the first BSNis a pure Gaussian state characterized by ( Γ , d ), the ele-ments of the QFIM are written as [29–34] H ij =2Tr[ Γ ( i,j )probe Γ ( j,i )probe ] − δ ij + ( Ω d ( i )probe ) T [ Γ − ] ( i,j ) × ( Ω d ( j )probe ) , (2)where A ( i,j ) denotes the 2 × i th rowand j th column of the M × M block matrix A , and similarfor the vector d ( i ) . The derivation of the QFIM of Eq. (2)is provided in Appendix A. The convexity of QFIM makesit sufficient to consider only pure probe states to find anoptimal state maximizing the QFIM [14], but one canfind the analytical form of the QFIM for general Gaus-sian states [29–34]. The quantum Cram´er-Rao bound in Eq. (1) can be saturated since the generators of param-eters commute [12]. B. Optimal product Gaussian state
Let us first consider the case where the probe state isa product state and thus the QFIM is evidently a diago-nal block matrix. Without loss of generality, we assumethat the block matrix of the covariance matrix for i thmode is Γ ( i,i ) = diag( e r i , e − r i ) /
2, simplifying the esti-mation error of φ ∗ to be ∆ φ ∗ ≥ (cid:80) Mi =1 w i / (cosh 4 r i − d i e − r i + 2 d i − e r i ).When probing with a product coherent state, the er-ror bound becomes (cid:80) Mi =1 w i / d i + d i − ), and the beststrategy for a given total average photon number ¯ N isto distribute the energy ¯ N over the modes according tothe weight | w i | , i.e., ¯ N i = ( d i + d i − ) / N | w i | . Theestimation error is thus∆ φ ∗ ≥ M (cid:88) i =1 w i N i = 14 ¯ N ≡ ∆ φ ∗ SQL , where the lower bound defines the standard quantumlimit. When w i = 1 /M , i.e., φ ∗ is the average phase,∆ φ ∗ SQL = 1 / M ¯ n , where ¯ n ≡ ¯ N /M represents an equalaverage number of photons hitting each phase shifter.Among all product Gaussian states, the best strategyunder the energy constraint ¯ N is to prepare the probestate in a product squeezed vacuum state with 8 ¯ N i ( ¯ N i +1) / (2 ¯ N i + 1) ∝ w i . Thus, particularly when w i = 1 /M ,in which φ ∗ is the average phase, the estimation errorbecomes∆ φ ∗ ≥ M N ( ¯ N + M ) = 18 M ¯ n (¯ n + 1) ≡ ∆ φ ∗ OPGS , (3)where we have set r i = r for all i and ¯ N = M sinh r .Note that the Heisenberg scaling with ¯ n or ¯ N is achieved.We refer to the above product squeezed vacuum state as the optimal product Gaussian state (OPGS) throughout (cid:1)(cid:2)(cid:3) (cid:1)(cid:2)(cid:4) (cid:3) (cid:4) (cid:3)(cid:1) (cid:4)(cid:1) (cid:3)(cid:1)(cid:1)(cid:3)(cid:1) - (cid:1) (cid:3)(cid:1) - (cid:2) (cid:1)(cid:2)(cid:1)(cid:1)(cid:3)(cid:1)(cid:2)(cid:1)(cid:3)(cid:1)(cid:1)(cid:2)(cid:3)(cid:1)(cid:1)(cid:3) ¯ N
C. Optimal entangled Gaussian state
We now turn to the case when the first BSN is config-ured to create mode correlation for an injected productinput state. In order to find the ultimate sensitivity indistributed sensing using Gaussian states and an opti-mal probe state, one can further develop the inequalityof Eq. (1) as∆ φ ∗ ≥ w T H − w ≥ | w | w T Hw = | w | ˆ G ∗ ) ψ ≥ | w | ψ (∆ ˆ G ∗ ) ψ , where ˆ G ∗ = (cid:80) Mi =1 w i ˆ a † i ˆ a i is the generator of φ ∗ [13].From now on, let us focus on the estimation of the aver-age phase, i.e., w i = 1 /M . Using a series of inequalities,we show that the error for the average phase estimationis given by (see Appendix B for the detail)∆ φ ∗ ≥
18 ¯ N ( ¯ N + 1) = 18 M ¯ n ( M ¯ n + 1) ≡ ∆ φ ∗ OEGS , We note that the ultimate error ∆ φ ∗ OEGS scaleswith ¯ N − or ¯ n − , and is smaller than the error ∆ φ ∗ OPGS .A similar scaling has been discussed in Ref. [14], but withdifferent quantification of the resource.We show that the ultimate error ∆ φ ∗ OEGS can beachieved by using symmetric Gaussian probe states withzero displacement. The covariance matrix of pure sym-metric Gaussian probe states can be written as a M × M partitioned matrix Γ probe with submatrices Γ ( i,i )probe =diag( γ , γ ) for all i and Γ ( i,j )probe = diag( (cid:15) , (cid:15) ) for all i (cid:54) = j [30, 36–38]. Since the states are assumed to be pure, thecomponents obey the relations ( γ − (cid:15) )( γ − (cid:15) ) = 1 / γ + ( M − (cid:15) ][ γ + ( M − (cid:15) ] = 1 / H ii = H for all i and H ij = H forall i (cid:54) = j . Finally, using Eqs. (1) and (2), the estimationerror is reduced to∆ φ ∗ ≥ M [ H + ( M − H ] , (4)where H = 2( γ + γ ) − H = 2( (cid:15) + (cid:15) ) (seeAppendix C for details). It is clear that the correlationquantified by (cid:15) and (cid:15) , H , plays an important role, but the sensitivity is eventually determined by an inter-play with the term H that is not independent of H for a given energy. After minimizing the lower bound inEq. (4) under the energy constraint ¯ N = M ( γ + γ − / φ ∗ OEGS when γ , = 1 / (cid:15) , and (cid:15) , = [ ¯ N ± (cid:112) ¯ N ( ¯ N + 1)] /M , leading to H = 4 ¯ N (2 ¯ N + M + 1) /M and H = 4 ¯ N (2 ¯ N + 1) /M . Therefore, the ultimate es-timation error ∆ φ ∗ OEGS can be achieved by the optimalsymmetric Gaussian state, which we call the optimal en-tangled Gaussian state (OEGS) throughout this paper.Most importantly, in contrast to the error ∆ φ ∗ OPGS , theerror ∆ φ ∗ OEGS is independent of the number of modes M for a fixed energy ¯ N and scales with M − for a fixed ¯ n ,evidently resulting from exploiting entanglement. Thus,the mode entanglement enables one to prevent the esti-mation error from growing with M . D. Role of entanglement
One might wonder whether the OEGS is the maxi-mally entangled Gaussian state, for which the entropy ofthe reduced state is maximized. We now demonstratethat it is not the case. The entropy of the single-modereduced state having a diagonal covariance matrix γ isgiven by S ( γ ) = ¯ n T ln(1 + 1 / ¯ n T ) + ln(¯ n T + 1) [21],where ¯ n T = √ γ γ − / S ( γ ) increases with the entanglement of the totalsystem under investigation, where pure symmetric Gaus-sian states are only considered [39]. Interestingly, theOEGS achieving the ultimate sensitivity does not havethe maximal entropy, as shown in Fig. 3(a). This is sur-prising and in contrast to other cases, where maximallyentangled states have shown to lead to the optimal sensi-tivity, e.g., the GHZ state of qubits exhibiting the maxi- ��� ��� ��� ��� ��������������������� (a)(b) Reduced entropy S ( )
FIG. 3. Dependence of the reduced entropy S ( γ ) in the es-timation error when probing with symmetric Gaussian states(black curves) in comparison with the OPGS (blue lines) andthe OEGS (red lines). (a) For the average phase estimation,the OEGS achieving the ultimate sensitivity does not have themaximal reduced entropy. (b) However, the OPGS is optimalfor the simultaneous phase estimation. mal entropy of the reduced state [13]. In our scenario, thestate often referred to as the continuous variable GHZ-type state having the maximal reduced entropy [40–42]exhibits worse sensitivity than the OEGS. A similar re-sult has been reported for estimation of unitarily gener-ated parameters in Ref. [10].It is worth comparing with the error of simultaneousphase estimation, ∆ φ ≡ (cid:80) Mi =1 ∆ φ i . For general sym-metric Gaussian states without displacement, the errorcan be written as ∆ φ ≥ Tr[ H − ] = ( M − / ( H − H ) + 1 / [ H + ( M − H ], where the first term willbe ignored if H = H . For a product probe state, H disappears and thus,∆ φ ≥ M N ( ¯ N + M ) ≡ ∆ φ OPGS , where the bound ∆ φ ∗ OPGS can be achieved by the OPGS.When using the OEGS, however, the estimation error isgiven by∆ φ ≥ M [2 ¯ N ( M −
1) + 2 M − N ( ¯ N + 1) ≡ ∆ φ OEGS . It is clear that the error ∆ φ OEGS is larger than the er-ror ∆ φ OPGS . More generally, any entangled symmetricGaussian states exhibit worse sensitivity than the OPGS,as shown in Fig. 3(b).
IV. PRACTICAL PERSPECTIVESA. Physical implementation of the optimal scheme
We have shown above that the ultimate estimationerror ∆ φ ∗ OEGS is achieved by the OEGS. Generationof the latter is experimentally feasible with currenttechnology as we provide here. Suppose that a prod-uct state of a p -squeezed vacuum and ( M −
1) vacuais injected into the first BSN, configured as ˆ U BSN =ˆ B M − ,M ( θ M − ) ˆ B M − ,M − ( θ M − ) × · · · × ˆ B , ( θ ), whereˆ B i,j ( θ j ) = exp[ θ j (ˆ a † i ˆ a j − ˆ a i ˆ a † j )] and θ j = arccos( M − j + 1) − / . Consequently, one can show that the out-put state of the BSN is the OEGS. Notice that differentconfigurations of BSN can be employed to generate theOEGS [16].We demonstrate here that homodyne detection on eachmode is sufficient to achieve the ultimate error ∆ φ ∗ OEGS without using the second BSN. The resultant probabil-ity distribution of homodyne detection follows a Gaus-sian distribution with the zero first-moment vector andthe M × M covariance matrix Γ HD with diagonal com-ponents [ Γ HD ] ii = γ cos ϕ i + γ sin ϕ i and off-diagonalcomponents [ Γ HD ] ij = (cid:15) cos ϕ i cos ϕ j + (cid:15) sin ϕ i sin ϕ j where ϕ i = φ i − θ HD ,i with homodyne angles θ HD ,i on i thmode. The error is thus given by ∆ φ ∗ HD ≥ w T F − w ,where F ij = Tr[ Γ − ( ∂ φ i Γ HD ) Γ − ( ∂ φ j Γ HD )] /
2. It can beeasily shown that the lower bound is equal to ∆ φ ∗ OEGS ��� ��� ��� ��� ��� ��������������������� (a) (b) ⌘
From a practical perspective, we analyze the effect ofphoton loss on the sensitivity. When loss is assumedto occur in each mode with an equal η , the covariancematrix of the probe state is transformed as Γ probe → η Γ probe + (1 − η ) M /
2, i.e., γ , → ηγ , + (1 − η ) / (cid:15) , → η(cid:15) , [21, 30]. Consequently, the theoreticaloptimal error bounds ∆ φ ∗ OPGS and ∆ φ ∗ OEGS become∆ φ ∗ OPGS ( η ) ≡ / N η (2 ¯
N η/M + η + 1) , ∆ φ ∗ OEGS ( η ) ≡ / N η (2 ¯
N η + η + 1) , respectively. When homodyne detection is performed,the resulting error bounds are respectively given as∆ φ ∗ OPGS,HD ( η ) ≡ [4 ¯ N η (1 − η ) + M ] / [8 η ¯ N ( ¯ N + M )] , ∆ φ ∗ OEGS,HD ( η ) ≡ [4 ¯ N η (1 − η ) + 1] / [8 η ¯ N ( ¯ N + 1)] , for which the homodyne angles have been appropriatelychosen. One may also seek other type of Gaussianmeasurement that could outperform the case yielding∆ φ ∗ OEGS,HD ( η ) in the presence of loss. We exemplify thelatter by performing an appropriate general-dyne detec-tion on the first output mode and heterodyne detectionon the other output modes of the second BSN that isset to realize ˆ U − . The associated error bound when � � � � � � ������������������� ✶ ✶ ✶ ✶ ✶ ✶○ ○ ○ ○ ○ ○△ △ △ △ △ △◻ ◻ ◻ ◻ ◻ ◻● ● ● ● ● ● ●▲ ▲ ▲ ▲ ▲ ▲ ▲■ ■ ■ ■ ■ ■ ■ � � � � � � ������ (a) (b) ¯ n
1. Comparable behaviorsbetween ∆ φ ∗ OEGS,HD ( η ) and ∆ φ ∗ OEGS,GD ( η ) are elab-orated in terms of ¯ N and η in Fig. 4(b), identifyingthe regions in which one prevails over the other. Itshows that homodyne detection is advantageous when¯ N > (1 + √ / η (1 − η ). Interestingly, the errorbound ∆ φ ∗ OEGS,HD ( η ) is exactly the same as that of asingle-mode phase estimation using a squeezed thermalstate [43]. One could further reduce the error by havingdisplacement as in Ref. [16], or seek for non-Gaussianmeasurements to achieve the ultimate error ∆ φ ∗ OEGS ( η )in lossy cases [32, 43].The enhancement of sensitivity by entanglement canbe quantified by the relative error ratio R opt =∆ φ ∗ OPGS ( η ) / ∆ φ ∗ OEGS ( η ) for the case that an optimalmeasurement is assumed, and the error ratio R HD =∆ φ ∗ OPGS,HD ( η ) / ∆ φ ∗ OEGS,HD ( η ) for the case that homo-dyne detection is performed. Figure 5(a) shows thatthe R opt slightly decreases with a moderate loss η andmonotonically increases with ¯ n , while the R HD drasti-cally drops with η and exhibits the optimum at ¯ n =1 / (cid:112) M η (1 − η ), where the relative enhancement is max-imal, when η <
1. The behaviors of R opt and R HD withincreasing M are presented in Fig. 5(b) for ¯ n = 6. Re-markably, both R opt and R HD are always greater thanunity in all cases with any η , stressing the usefulness of entanglement in Gaussian distributed sensing againstloss. V. DISCUSSION
We have investigated the ultimate sensitivity of theaverage phase estimation in distributed quantum sensingusing Gaussian states. The ultimate sensitivity has beenshown to be achievable by the OEGS possessing partialentanglement between the modes and by performing ho-modyne detection on each mode in the absence of loss.When photon loss occurs, homodyne detection ceases tobe optimal, but non-Gaussian measurement would be re-quired for achieving the ultimate sensitivity. Alterna-tively, a slightly better sensitivity can be obtained byconducting other type of Gaussian measurement on theoutput modes of the second BSN that implements theinverse transformation of the first BSN. Although thesensitivity decreases with loss in all the cases consideredin this work, we have revealed that using the OEGS is al-ways advantageous for average phase estimation as com-pared to the case using unentangled symmetric Gaussianstates. While we have focused on identification of the ul-timate sensitivity and the optimal setup for the averagephase estimation in this work, finding those for estima-tion of other linear combinations of phases would alsobe an interesting future study. Another interesting openquestion is to explain the enhancement of sensitivity in amore intuitive manner such as using the multiparametersqueezing parameter [18] other than entanglement, whichwe leave for future study. Finally, finding the optimalmeasurement achieving the best sensitivity in the pres-ence of photon-loss is also an important remaining taskas in the recent study, where the optimal non-Gaussianmeasurement in a special case of M = 1 is theoreticallyfound [43]. The experimental implementation of the op-timal measurement needs to be devised.It is worthwhile to discuss our results in relation torecent results in distributed sensing. First of all, arecent experiment successfully showed an enhancementby entanglement in distributed Gaussian quantum sens-ing [16]. The theory behind the experiment in Ref. [16]assumed that the phase shifts of interest were extremelysmall and the estimation error was quantified by the lin-ear error propagation analysis from homodyne detection.However, our work identifies the ultimate estimation er-ror in distributed Gaussian sensing by proposing the op-timal Gaussian probe and it can be applied to phaseshifts of arbitrary degrees. Thus, the experimental re-sults could be understood better and interpreted from abroader perspective of distributed Gaussian sensing. Inaddition, the optimal entangled Gaussian state has beenproven to be optimal for distributed quantum sensing offield-quadrature displacement [44–46].This work was supported by National Research Foun-dation of Korea (NRF) grants funded by the Korea gov-ernment (Grants No. NRF-2019R1H1A3079890 and No.NRF-2018K2A9A1A06069933).C.O. and C.L. contributed equally to this work. Appendix A: Derivation of the quantum Fisher information matrix for distributed sensing using isothermalGaussian states
In this section, we derive the quantum Fisher information matrix (QFIM) for distributed sensing using isothermalGaussian states. When a phase-encoded state is the isothermal Gaussian quantum states characterized by [ Γ ( φ ) , d ( φ )]with a isothermal photon number ¯ n , the QFIM is given by [32] H ij = 12¯ n + 2¯ n + 1 Tr (cid:20) Ω M ∂ Γ ( φ ) ∂φ i Ω M ∂ Γ ( φ ) ∂φ j (cid:21) + ∂ d T ( φ ) ∂φ i Γ − ∂ d ( φ ) ∂φ j , (A1)where Γ ( φ ) = S ( φ ) Γ probe S T ( φ ) , d ( φ ) = S ( φ ) d probe , are the covariance matrix and the first moment vector of the quantum state after the unitary operation encoding φ cor-responding to the symplectic matrix S ( φ ), respectively. In the distributed phase sensor, the symplectic transformationcorresponds to S ( φ ) = ⊕ Mi =1 (cid:18) cos φ i sin φ i − sin φ i cos φ i (cid:19) . Note that symplectic transformation S is defined as ones that preserve the canonical commutation relation, S T Ω M S = Ω M , corresponding to a Gaussian unitary operation ˆ U applied to density matrices by the relation ˆ U † ˆ Q ˆ U = S ˆ Q .The first term in Eq. (A1) can be simplified asTr (cid:20) Ω M ∂ Γ ∂φ i Ω M ∂ Γ ∂φ j (cid:21) =Tr (cid:20) Ω M ∂S ( φ ) ∂φ i Γ probe Ω M ∂S ( φ ) ∂φ j Γ probe + Ω M ∂S ( φ ) ∂φ i Γ probe Ω M Γ probe ∂S T ( φ ) ∂φ j + Ω M Γ probe ∂S T ( φ ) ∂φ i Ω M ∂S ( φ ) ∂φ j Γ probe + Ω M Γ probe ∂S T ( φ ) ∂φ i Ω M Γ probe ∂S T ( φ ) ∂φ j (cid:21) φ =0 = Tr[ P i Γ probe P j Γ probe + P i Γ probe Ω M Γ probe Ω M P j + P j Γ probe Ω M Γ probe Ω M P i + Γ probe P i Γ probe P j ]=2Tr[ Γ ( i,j )probe Γ ( j,i )probe ] − δ ij (2¯ n + 1) (A2)where Γ ( i,j )probe = P i Γ probe P j . Here, we have set φ = 0 without loss of generality since the QFIM is independent of φ under unitary transformation, and we have used ∂ Γ ( φ ) ∂φ i = ∂S ( φ ) ∂φ i Γ probe S T ( φ ) + S ( φ ) Γ probe ∂S T ( φ ) ∂φ i = ∂S ( φ ) ∂φ i Γ probe + Γ probe ∂S T ( φ ) ∂φ i , and − Ω M ∂S ( φ ) ∂φ i = − ∂S ( φ ) ∂φ i Ω M = Ω M ∂S T ( φ ) ∂φ i = ∂S T ( φ ) ∂φ i Ω M , which is the projection onto the i th mode, P i = − Ω M ∂S ( φ ) ∂φ i (cid:12)(cid:12)(cid:12)(cid:12) φ =0 when φ = 0.The second term in Eq. (A1) is ∂ d T ( φ ) ∂φ i Γ − ∂ d ( φ ) ∂φ j = (cid:18) ∂S ( φ ) ∂φ i d probe (cid:19) T Γ − (cid:18) ∂S ( φ ) ∂φ j d probe (cid:19) φ =0 = ( Ω d ( i )probe ) T [ Γ − ] ( i,j ) ( Ω d ( i )probe ) . (A3)Thus, substituting ¯ n = 0 into Eqs. (A1) ∼ (A3), i.e., for pure states and using Γ probe = Γ and d probe = d since wehave set φ = 0, we obtain the expression of Eq. (2) in the main text. Appendix B: Maximum variance of ˆ G ∗ Let us derive the maximum variance of ˆ G ∗ = (cid:80) Mi =1 ˆ a † i ˆ a i /M . The variance can be written as M (∆ ˆ G ∗ ) = M ( (cid:104) ˆ G ∗ (cid:105) − (cid:104) ˆ G ∗ (cid:105) ) = (cid:104) (cid:32) M (cid:88) i =1 ˆ a † i ˆ a i (cid:33) (cid:105) − (cid:32) M (cid:88) i =1 (cid:104) ˆ a † i ˆ a i (cid:105) (cid:33) = M (cid:88) i =1 (cid:104) (ˆ a † i ˆ a i ) (cid:105) + M (cid:88) i (cid:54) = j (cid:104) ˆ a † i ˆ a i ˆ a † j ˆ a j (cid:105) − M (cid:88) i =1 (cid:104) ˆ a † i ˆ a i (cid:105) − M (cid:88) i (cid:54) = j (cid:104) ˆ a † i ˆ a i (cid:105)(cid:104) ˆ a † j ˆ a j (cid:105) . Using the fact that ˆ G ∗ is invariant under any passive transformation, one can assume that (cid:104) ˆ a † i ˆ a i ˆ a † j ˆ a j (cid:105)−(cid:104) ˆ a † i ˆ a i (cid:105)(cid:104) ˆ a † j ˆ a j (cid:105) = 0for i (cid:54) = j without loss of generality and get ∆ ˆ G ∗ = 1 M M (cid:88) i =1 ∆ (ˆ a † i ˆ a i ) , which shows the variance of ∆ ˆ G ∗ is the sum of the photon number variance in all the modes. Since a squeezedvacuum state exhibits the maximum photon number variance among Gaussian states, which is (cosh 4 r − / ˆ G ∗ ≤ M M (cid:88) i =1 (cosh 4 r i − . (B1)Under the constraint for the total mean photon number of the state ¯ N , one can prove that the upper bound of ∆ ˆ G ∗ in Eq. (B1) is given by 2 ¯ N ( ¯ N + 1), i.e., 4∆ ˆ G ∗ ≤ N ( ¯ N + 1) M . (B2) Appendix C: Properties of the QFIM for symmetric Gaussian states
Let us consider the QFIM having diagonal elements H and off-diagonal elements H , which is then written as H = H M (cid:88) i =1 | i (cid:105)(cid:104) i | + H M (cid:88) i (cid:54) = j | i (cid:105)(cid:104) j | = ( H − H ) M (cid:88) i =1 | i (cid:105)(cid:104) i | + H M (cid:88) i,j =1 | i (cid:105)(cid:104) j | = ( H − H ) + H M (cid:88) i,j =1 | i (cid:105)(cid:104) j | , where {| i (cid:105)} Mi =1 represents the standard basis. By introducing | + (cid:105) = (cid:80) Mi =1 | i (cid:105) / √ M , H = ( H − H ) + M H | + (cid:105)(cid:104) + | = ( H − H )( − | + (cid:105)(cid:104) + | ) + [( M − H + H ] | + (cid:105)(cid:104) + | . When H (cid:54) = H , the inverse of the QFIM is H − = ( H − H ) − ( − | + (cid:105)(cid:104) + | ) + [( M − H + H ] − | + (cid:105)(cid:104) + | , and thus Tr[ H − ] = ( M − H − H ) − + [( M − H + H ] − . When H = H , however, the inverse of the QFIM is H − = [( M − H + H ] − | + (cid:105)(cid:104) + | , and thus Tr[ H − ] = [( M − H + H ] − . Appendix D: Minimization of the estimation error when probing with symmetric Gaussian states
For pure symmetric Gaussian states, the elements of the covariance matrix satisfy( γ − (cid:15) )( γ − (cid:15) ) = 1 / , [ γ + ( M − (cid:15) ][ γ + ( M − (cid:15) ] = 1 / , and the energy constraint is ¯ N = M ( γ + γ − / . Parametrizing γ , as γ , = ¯ n T e ± r , we can rewrite (cid:15) , as (cid:15) , = 2 + 4¯ n ( M − − M ± (cid:112) (4¯ n − M (4¯ n M − M + 4) − n T ( M − e ± r , where 0 ≤ r ≤ cosh − (2 ¯ N /M + 1) / M [2( γ + γ ) − M − (cid:15) + (cid:15) )] , (D1)whose maximum value can be shown to be 8 ¯ N ( ¯ N + 1) in general.One can easily check that if we use γ , = 12 + ¯ N ± (cid:112) ¯ N ( ¯ N + 1) M , (D2)and (cid:15) , = ¯ N ± (cid:112) ¯ N ( ¯ N + 1) M , (D3)the maximum value of Eq. (D1), i.e., 8 ¯ N ( ¯ N + 1), is attained, which thus proves that ∆ φ ∗ OEGS introduced in the maintext is achievable by the symmetric Gaussian states with parameters satisfying Eqs. (D2) and (D3).
Appendix E: General Gaussian measurement
In this section, we derive the lower bound of the estimation error based on a particular Gaussian measurement andprovide its implementation. A measurement is called a Gaussian measurement if it can be implemented by addingGaussian ancilla states with Gaussian unitary operations and performing homodyne detection [22, 30]. Mathemat-ically, a Gaussian measurement on M -mode states ˆ ρ can be written by positive-valued measure measure (POVM)elements { ˆΠ ξ } as ˆΠ ξ = 1 π M ˆ D ( ξ ) ˆΠ ˆ D † ( ξ ) , where ˆ D ( ξ ) = exp( − i ξ T Ω M ˆ Q ) is a displacement operator, and ˆΠ is a density matrix of a M -mode Gaussian statewith a zero-displacement and a covariance matrix Γ M . Note here that ˆΠ characterizes the Gaussian measurement.Let us assume ˆΠ to be a pure state. One can easily show that the probability distribution for a Gaussian input statewith the covariance matrix Γ and the first moment d is given as a Gaussian distribution with the covariance matrix( Γ + Γ M ) / d / √
2. For the phase-encoded Gaussian state of Γ( φ ) with zero displacement, theFisher information elements based on Gaussian measurement with Γ M are thus given by F ij ( φ ) = 12 Tr (cid:20) ( Γ + Γ M ) − ∂ Γ ∂φ i ( Γ + Γ M ) − ∂ Γ ∂φ j (cid:21) . Let us consider a Gaussian measurement ˆΠ with the following covariance matrix: Γ M = γ M (cid:15) M ... (cid:15) M (cid:15) M γ M ... (cid:15) M ... ... ... ... (cid:15) M (cid:15) M ... γ M , where γ M = diag( γ M , , γ M , ) and (cid:15) M = diag( (cid:15) M , , (cid:15) M , ) are 2 × γ M ,j = 1 / (cid:15) M ,j and (cid:15) M ,j = [ ¯ N M − ( − j (cid:112) ¯ N M ( ¯ N M + 1)] /M for j = 1 ,
2. Note that the covariance matrix is the same as that of theoptimal entangled Gaussian state with ¯ N replaced by ¯ N M . If the phase-encoded state is the optimal entangledGaussian state in the presence of loss, then one can find that the lower bound of the error can be written as∆ φ ∗ ≥ η ¯ N M ¯ N − η (cid:112) ¯ N M ( ¯ N M + 1) ¯ N ( ¯ N + 1) + ¯ N M − ( η − η ¯ N + 14 η ¯ N ( ¯ N + 1) ≥ N (1 − η ) η + 1 + (cid:112) − N η ( η − η ¯ N ( ¯ N + 1) , where the optimal value of ¯ N M is chosen for the second inequality.If we employ a squeezed thermal input state, ˆ ρ in = ˆ S ( r )ˆ ρ T ˆ S † ( r ) ⊗ | (cid:105)(cid:104) | ⊗ M − where ˆ S ( r ) = exp[ r (ˆ a † − ˆ a ) /
2] is asqueezing operator applied on the first mode, and ˆ ρ T = (cid:80) ∞ n =0 ¯ n n / (¯ n + 1) n +1 | n (cid:105)(cid:104) n | is a thermal state with the meanphoton number ¯ n , the lower bound by the aforementioned Gaussian measurement can be written as∆ φ ∗ ≥ (cid:34)(cid:18) n + 1¯ n + 1 (cid:19) sinh r (cid:35) − , which is exactly the same as the lower bound for single-mode phase estimation using a squeezed thermal probe state, asshown in Ref. [43]. Note that preparing the squeezed thermal state input without a photon-loss channel is equivalentto using the optimal entangled Gaussian state with a photon-loss channel after adjusting appropriate parameterswhen the photon-loss rates are equal to each other [47].Let us find the implementation of the Gaussian measurement corresponding to Γ M . Noticing that mixing a p -squeezed state and ( M −
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