Optimal Gaussian metrology for generic multimode interferometric circuit

Bounds on the ultimate precision attainable in the estimation of a parameter in Gaussian quantum metrology are obtained when the average number of bosonic probes is fixed. We identify the optimal input probe state among generic (mixed in general) Gaussian states with a fixed average number of probe photons for the estimation of a parameter contained in a generic multimode interferometric optical circuit, namely, a passive linear circuit preserving the total number of photons. The optimal Gaussian input state is essentially a single-mode squeezed vacuum, and the ultimate precision is achieved by a homodyne measurement on the single mode. We also reveal the best strategy for the estimation when we are given L identical target circuits and are allowed to apply passive linear controls in between with an arbitrary number of ancilla modes introduced.

In a variety of quantum optical metrology settings, the probe sensitivity to the target parameter can be improved by squeezing the state of the input light [7,8]. Entanglement is also an important keyword in the studies of quantum metrology [4][5][6]. In these ways, the state of the input probe photons is important for high precision metrology.
For instance, the estimation of a single-mode phase shift is studied with pure [14] and mixed [19] Gaussian probes, and some other single-mode Gaussian channels such as squeezing and amplitude-damping are analyzed with general mixed Gaussian probes [34]. The estimation of a single-mode phase shift with general mixed Gaussian probes is discussed in the presence of general Gaussian dissipation [60]. A few specific two-mode Gaussian channels like twomode squeezing and mode mixing are studied with some particular types of two-mode Gaussian probes [59]. The ultimate precision bound is clarified for generic two-mode passive linear circuits, which preserve the number of photons passing through them (they are Gaussian channels) [47]. A formula for the quantum Fisher information (QFI) valid for any multimode pure Gaussian states is derived and investigated under the condition of intense probe light (with large displacement) [25]. General multimode Gaussian unitary channels (Bogoliubov transformations) are considered with pure probe states not restricted to Gaussian states and the behavior of the QFI for large mean photon numbers is discussed [46]. A formula for the QFI matrix is derived for general multimode Gaussian states and multiparameter Gaussian quantum metrology is discussed [61,64].
In this paper, we study the estimation of a parameter embedded in a generic M-mode passive linear interferometric circuit, and clarify the ultimate precision bound achievable with Gaussian probes. We identify the optimal input probe state among all Gaussian states (including mixed Gaussian states) with a fixed average number of probe photons. Such a bound is known for M=2 [47], but is not known for M3. The proof strategy taken for M=2 is not helpful for M3, and it is not a simple generalization of the previous work.
More specifically, we will consider the setting shown in figure 1: a collection of M photonic modes is employed as a probe to recover the value of an unknown parameter j, which is imprinted on the state of the probe via the action of a passive (i.e. photon number preserving), Gaussian (i.e. mapping Gaussian input into Gaussian output), unitary transformation U ĵ . Under the assumption that the allowed input density matrices r of the M modes belong to the set M N , ( )of (not necessarily pure) Gaussian states with an average photon number N , we are interested in the ultimate accuracy in the estimation of j attainable when having full access to the output state To this end, we shall focus on the QFI F j r ( |ˆ) of the problem, which, via the quantum Cramér-Rao inequality [1,4,[85][86][87][88][89][90], sets a universal bound on min dj that is independent of the adopted measurement procedure, Here, g j  is the spectral norm of the Hermitian matrix  (3.31). We note that, apart from some special cases, such optimal vectors opt y ñ | generally depend on the variable j, whose unknown value we wish to determine. Therefore, the possibility of using this optimal input state for achieving the bound is not straightforward, and would require in practice the use of iterative procedures with a sequence of input states that approximate the optimal state. Anyway, the optimal state opt y ñ | enables us to reach the upper bound(1.4).
The paper is organized as follows. The model and the estimation problem are set up in section 2. In section 3, the maximal precision achievable by a Gaussian probe is found, first for pure Gaussian states and then for mixed Gaussian states. Moreover, we explicitly find the optimal states that achieve the maximal precision. Two different measurement schemes are presented in section 4. We look at a few simple examples in section 5. Figure 1. The generic passive linear optical circuit U ĵ with M input ports and M output ports. Our problem is to estimate a parameter j contained in the circuit U ĵ , by sending probe photons through it and measuring its output. We will restrict ourselves to Gaussian input states r with a given average number of probe photons N N á ñ = , among which we identify the best Gaussian states reducing the accuracy limit in the estimation of j as much as possible.
Furthermore, in section 6, we exhibit the optimal sequential strategy for the estimation when several target circuits, together with ancilla modes, are allowed to be used. A summary of the present work is given in section 7. We add four appendices, containing some technical tools and proofs. In appendix A we collect some results on Gaussian states and operations, in appendix B we show the derivation of a formula for the QFI, in appendix C we prove some inequalities on Hermitian matrices used in the solution of the optimization problems, and appendix D contains the proof of the optimality of the measurement scheme presented in section 4.

The model
where U j is an M×M unitary matrix, whose functional dependence upon j is assumed to be smooth. We remind that this kind of transformation preserves the total number of photons of the system, i.e. This last condition is motivated by the fact that it is not realistic to consider probing signals with unbounded input energy. It turns out that for generic (non-Gaussian) input states the constraint(2.10) is not strong enough to keep the QFI F j r ( |ˆ) finite (see for instance [47], where, for the case with M = 2 input modes, obtaining finite optimal values for F j r ( |ˆ) requires to impose an extra condition on the variance of N on ; r see also [27]), yet for Gaussian inputs this suffices and the QFI F j r ( |ˆ) is finite under the constraint(2.10).   denotes the expectation value on r . Furthermore, since U ĵ is a passive Gaussian unitary, the associated output state r ĵ obtained as(1.1) also belongs to M N , ( ), and its covariance matrix G j and displacement vector d j depend linearly on Γ and d, as

Optimization of QFI
where R j is the orthogonal matrix rotating the quadrature operators according to U ĵ (see appendix A). Under this condition, the SLD fulfilling(2.9) can be expressed as [29] and, accordingly, the QFI reads [29,93] known as the symplectic form.
In the remainder of this section, we shall employ these expressions to derive the inequality(1.4). The analysis will be split into two parts, addressing first the case of the pure elements of M N , ( )and then the case of the mixed ones. For those who are familiar with QFI optimization problems, this procedure might sound unnecessary. Indeed, due to the convexity of QFI [4,94], it is well-known that pure input states perform better than mixed input states for metrological purposes. We cannot, however, apply the same argument in the present case, and it is not obvious at first glance whether the best state is a pure state. Indeed, even though it is true that any mixed Gaussian state can be decomposed as a convex sum of pure Gaussian states, each of the constituent of such decomposition does not necessarily satisfy the constraint(2.10) on the photon number in general. In short, the Gaussian set M N , ( )is not a convex set, and therefore we cannot use the convexity argument to optimize the QFI. As a consequence, for the problem we are considering here, we have to address explicitly the case of mixed input states.

Optimization among pure Gaussian inputs
with d being the displacement vector of yñ | . Exactly the same properties hold for the covariance matrix G j and the displacement d j of the associated output counterpart(1.1) of yñ | , which of course is also a pure element of the set M N , ( ). Under this premise, the equation (3.7) for L j can be solved explicitly, yielding with the SLD (3.5) given by A further simplification can then be obtained by invoking (3.4), which expresses the functional dependence of G j and d j in terms of the symplectic orthogonal matrix R j representing the passive Gaussian unitary transformation U ĵ . Specifically, we get is the generator of R j . Our problem is, therefore, to maximize the QFI F j r ( |ˆ) in(3.14) with respect to Γ and d, keeping in mind the parameterization(3.9) and the constraint(3.10). For this purpose, we start bounding the first term F 1 j r ( |ˆ) ( ) in the sum(3.14). By plugging the symplectic decomposition(3.9) of Γ, and using the parameterization(A.12) for R as well as the structure(A.21) of the generator G j , we get (see appendix B for the derivation) Tr cosh 2 Tr sinh 2 sinh 2 Tr 3.17 where g j is the generator of the unitary matrix U j as introduced in (1.5) and involved in the structure of G j in (A.21), while U is the unitary matrix appearing in the parameterization of R in (A.12). This quantity can be bounded from above as where we have used the inequalities valid for Hermitian matrices A and B, and valid for Hermitian and positive semi-definite matrices A and B (see appendix C for their proofs). Note that g j is Hermitian and hence U g U 2 j ( ) † is positive semi-definite, and its norm is given by U g U g , on the other hand, can be bounded from above as where we have assumed, without loss of generality, that r 0 m  (m=1, K, M). Exploiting these results, we can then bound the QFI(3.14) as where we have used the inequality valid for a positive semi-definite matrix A, which is saturated if and only if only one of the eigenvalues of A is nonvanishing and it is not degenerate (see appendix C for its proof). Imposing hence the constraint(3.10), this finally gives us which proves the inequality(1.4) for the case of pure input Gaussian states. This result reproduces the bounds previously known for M=1 (single-mode phase shift) [14,19,34,40,59] and for M=2 (general two-mode passive linear circuits) [47], and generalizes them to M 3  .

Optimal states
The above derivation of the bound not only proves that the inequality(1.4) holds at least for the pure input states of the set M N , ( ), but also that the bound is saturated by a proper choice of the inputs, i.e.by properly tuning the parameters in Γ and d. Let us identify such input states.
(i) In order to saturate the last inequality in(3.25), the necessary and sufficient condition is are satisfied: recall the conditions for the equalities in (3.19) and (3.20). These conditions are already satisfied with the above tunings of r and U in(3.27) and(3.28).
the squeezing operator on the first mode. A couple of comments are in order. First, recall that V ĵ is the passive linear transformation characterized by with the M×M unitary matrix V j diagonalizing the generator g j of the circuit as in(A.23). It redefines the modes of the system in a way that allows us to describe the optimal state as a configuration with all the photons injected into the first mode only (i.e. the one with the largest (in magnitude) eigenvalue of g j ). We stress, however, that even after this 'reorganization' the modes other than the first one are not free from the target parameter j in general, due to the subsequent propagation induced by U ĵ , and the problem is not reduced to a single-mode problem. It remains intrinsically a multimode problem, and we cannot simply apply the results known for single-mode estimation problems. Second, as indicated by the notation, the transformation V ĵ may depend upon the target parameter j for a generic choice of U ĵ , and so may do the optimal state opt y ñ | . Therefore, if that is the case, it would not be easy to prepare this optimal state opt y ñ | without knowing the value of the parameter j, which we intend to estimate, and an adaptive strategy updating the estimate of j would be required in practice.

Optimization among mixed Gaussian inputs
We have just shown that the inequality(1.4) holds at least for the pure elements of the set M N , ( ). Here, we are going to generalize this by showing that the same result holds for the mixed elements of the set M N , ( ). We first point out that any mixed Gaussian state d , r Ĝ , characterized by a covariance matrix Γ and a displacement d, can be expressed as a mixture of pure Gaussian states which is equivalent to (3.32 For the first equality, we have used the moments of the Gaussian distribution P

Measurements
In this section, we focus on the measurement  that attains the maximum on the right-hand side of(2.7) yielding the QFI. As it is the case for the optimal input state opt y ñ | analyzed in the previous section, we shall see that the optimal POVM also exhibits in general a nontrivial dependence on the target parameter j, making it problematic to use it in realistic situations. Still, determining the optimal POVM explicitly is a well-defined problem which deserves to be addressed.
As a starting point of our study, we use the well-known fact that a POVM  that maximizes the FI of the problem can always be constructed by looking at the set of the eigenprojections of the SLD L j of the model [89]. We have given an SLD L j for a generic Gaussian state r ĵ in(3.5), which for a pure Gaussian state reduces to (3.13). For our problem, in which the parameter j is embedded in the probe state via a passive linear circuit, it reduces further to(3.15), which depends on the input state, i.e.its covariance matrix Γ and displacement d, and the generator G j of the circuit. Specifying this expression in the case of the optimal input opt y ñ | in(3.31), we get with a 1 being the annihilation operator of the first probing mode, and 1 e being the largest (in magnitude) eigenvalue of G j , which is put in the first mode after the diagonalization of G j by V ĵ (see (A.21)-(A.24); recall also the discussion around (3.28)).
Notice, however, that SLD is not unique when the density operator r ĵ is not of full rank: see(2.9). Indeed, there is a different and simple construction of SLD for a pure state. Since a pure state r ĵ satisfies 2 r r = j ĵˆ, its derivative yields an SLD L 2d dr j = j j ¢ / , which for our problem with the optimal Gaussian input state opt y ñ | reads is the generator of the target circuit U ĵ , which is quadratic in the canonical operators â and â † . This SLD L ¢ ĵ is of rank 2, and its eigenbasis includes the two orthogonal eigenvectors will achieve the upper bound of the QFI in(1.4). This is a generalization of the result given in [14], from a singlemode phase shift to a generic multimode passive linear circuit. Another example of an optimal POVM can be obtained by considering the scheme depicted in figure 2 (the circuit in figure 2 includes both the preparation stage for the optimal input state opt y ñ | in (3.31) and the probing stage together with the circuit U ĵ ). The measurement is to first undo the circuit U ĵ as well as the transformation V ĵ applied to prepare the optimal input state opt y ñ | in(3.31), and then to perform the homodyne measurement on the first mode along the quadrature x x x y e e cos sin a a a a 1 i 1 i 1 1 ) . Indeed, the FI by this POVM x P {ˆ} for the optimal input opt y ñ | in(3.31) coincides with the upper bound of the QFI in(1.4). See appendix D for the proof. This is a generalization of the result given in [19], from a single-mode phase shift to a generic multimode passive linear circuit.     Notice that, in this case, the optimal input state opt y ñ | in(5.9) is independent of the target parameter j. Note also that the same expression as(5.10) is found e.g.in [14,19,34,40,59], but it is found there as the optimal QFI for the estimation of the single-mode phase shift with a Gaussian probe. Here, (5.10) is presented as the optimal QFI for the two-mode circuit in figure 3(a).

Simple examples
The

MZ interferometer II
Let us look at the MZ interferometer in the slightly different configuration shown in figure 3

/ /
We thus have The optimal Gaussian input state for this MZ interferometer is the same as the one given in(5.9), while the maximal QFI achievable by the optimal input state is This QFI is lower than the previous one in(5.10) for the other MZ interferometer, even though the relative phases j to be estimated in the two MZ interferometers are the same. This is because injecting all the resources to one of the two arms of the interferometer is optimal if we stick to Gaussian probes, and only one of the two phase shifters in figure 3(b) is probed. It would be worth noticing that our estimation problem implicitly assumes the presence of an external phase reference. Without the reference beam, the two MZ interferometers in figures 3(a) and (b) are equivalent, since only the relative phase between the two arms matters in such a case. See the discussion in [26].

Two-mode mixing
Let us look at another two-mode example: the estimation of the parameter j characterizing the transmissivity of the beam splitter represented by the unitary transformation figure 3(c)

= -
It is unitarily equivalent to the generator G ĵ in(5.12), apart from the numerical proportionality constant 1/2. We thus have and the maximal QFI is given by with the squeezing parameter r 0 given in (3.30). This optimal state is again independent of the target parameter j. The same estimation problem, i.e.the estimation of j in the two-mode mixing channel (5.17), is studied in [59], but the maximal QFI (5.21) and the optimal Gaussian input state (5.22) are not identified there.

Three-mode mixing
Let us also look at a three-mode example. We consider the circuit shown in figure 3(d), composed of two beam splitters of the same transmissivity characterized by the parameter j. Our problem is to estimate the single parameter j in the three-mode mixing circuit represented by the unitary transformation with the squeezing parameter r 0 given in (3.30). In this case, the optimal input state depends on the target parameter j. If our guess j¢ is not precise and does not match the true value j, the input state (3.31) and the measurement, e.g.(4.6) or (4.7), prepared and performed with the guessed value j¢ in place of j (see e.g. the circuit in figure 2) are not optimal, and the FI for such a nonoptimal probing deviates from the maximal QFI in(1.4). Since we assume that the functional dependence of U ĵ upon j is smooth, the FI is a smooth function of j¢, and therefore, the deviation of FI from the maximal QFI is only quadratic around the optimal point j j ¢ = .
In this sense, the FI is robust to a small error in the guess of j.

Sequential strategy
If we are allowed to use multiple (identical) target circuits U ĵ at the same time, we could do better. Suppose that we are given L identical M-mode passive linear circuits U ĵ . A paradigmatic scheme for the quantum metrology is the parallel scheme in figure 4(a) with an entangled input r [2,4]. The result in section 3 suggests, however, that, if we stick to Gaussian inputs, this parallel setup does not help improve the maximal QFI found in(1.4), since the best strategy is to inject all the resources into a single-mode of the overall LM-mode passive linear circuit in )modes in total in the overall circuit, and the unitary operators U ĵ act only on the first M modes, i.e.U  Ä ĵ . By abuse of notation, U  Ä ĵ is simply denoted by U ĵ in(6.1). The overall circuit is a K-mode passive linear circuit, and the orthogonal matrix  j which rotates the quadrature operators zˆin phase space according to the transformation  ĵ is given by ) are K×K unitary matrices corresponding to U  Ä ĵ and Û ℓ , respectively. The quantity relevant to the maximal QFI is the spectral norm of the generator of this orthogonal transformation  j (see (1.4)), i.e.the largest (in magnitude) eigenvalue of The spectral norm of the generator  j is bounded from above as A sufficient and general solution is given by Figure 4. The parallel scheme in (a) is equivalent to the sequential scheme in (b) with the target circuits U ĵ swapped by SWAP gates, which is a particular case of the sequential scheme in (c) with generic gates Û ℓ entangling the main probes with additional ancillas.
(see [95]). By this choice, the generator of the overall circuit  ĵ is reduced to Lg  = j j , and the upper bound on the QFI by the sequential strategy with a Gaussian input K N ,  r Î ( )is given by This upper bound is saturated by the input state 0 , 6.10 opt opt given in (3.31) for the first M modes while vacuum for the rest. The results in(6.8) and (6.10) show that the ancilla modes are not necessary for the optimal strategy. We note that in general the optimal controls(6.8) and the optimal input state(6.10) depend on the target parameter j.

Summary
We have clarified the universal bound(1.4) on the precision of the estimation (QFI) of a parameter embedded in a generic multimode passive (photon number preserving) linear optical circuit by using Gaussian probes with a given average number of probe photons N . We have identified the input Gaussian state(3.31) that yields the QFI saturating the bound(1.4): it is a single-mode squeezed vacuum in an appropriate basis. We have also found measurements (POVMs)(4.6) and(4.7) by which FI reaches QFI. The best (sequential) strategy when we are given multiple identical target circuits and are allowed to apply passive linear controls in between with the help of an arbitrary number of ancilla modes has been revealed: no ancilla mode is actually needed for the best strategy 6 .
Even though the optimal input state(3.31) and the optimal measurements(4.6) and(4.7), as well as the optimal controls(6.8) in the sequential strategy, depend on the target parameter to be estimated in general and adaptive adjustments of the input, the measurement, and the controls would be required to achieve the precision bound in practice, the above result shows that the bound is sharp and covers various specific setups composed of phase shifters and beam splitters, including the standard MZ interferometer, providing the universal bound that cannot be beaten by any Gaussian inputs and any passive controls.
The present work has focussed on passive linear circuits. Bounds on more general Gaussian metrology, for general Gaussian channels including amplitude-damping channels and channels involving squeezing, etc., have not been thoroughly understood yet, beyond analyses on specific setups. Entanglement with ancilla modes would be useful for such generic Gaussian metrology [17] and it would be interesting to explore.
There are works in the literature which discuss the unnecessity of mode entanglement [5,39,40,50,53,55,96,97]. Note, however, that in those works the probe states are not restricted to Gaussian states and in addition just the achievability of the Heisenberg scaling (quadratic in N ) is discussed. The chosen probe states are not necessarily the optimal ones, even though they actually yields QFIs scaling quadratically in N (their coefficients are not necessarily the optimal). On the other hand, in the present work, we look at the optimal state which yields the maximal QFI.
Aligning these operators as a column vector where á ñ  denotes the expectation value on r . In particular, its characteristic function reads as e e . A .8    As is clear from this structure, the matrix R j is symplectic and orthogonal, and the passive linear transformation U ĵ is a rotation on the phase space. By construction the transformation U ĵ maps the set M N , ( )into itself. In particular, given M N ,  r Î ( ), the covariance matrix G j and the displacement d j of the associated Gaussian output state r ĵ in(1.1) are obtained by rotating the covariance matrix Γ and the displacement d of the input state r as Note that they still fulfill the constraint(A.9) due to the fact that R j is orthogonal. An important role on our problem is played by the generator of the transformation U ĵ , i.e.by the operator