Abstract
Quantum metrology is the stateoftheart measurement technology. It uses quantum resources to enhance the sensitivity of phase estimation over that achievable by classical physics. While single parameter estimation theory has been widely investigated, much less is known about the simultaneous estimation of multiple phases, which finds key applications in imaging and sensing. In this manuscript we provide conditions of useful particle (qudit) entanglement for multiphase estimation and adapt them to multiarm MachZehnder interferometry. We theoretically discuss benchmark multimode Fock states containing useful qudit entanglement and overcoming the sensitivity of separable qudit states in three and four arm MachZehnderlike interferometers  currently within the reach of integrated photonics technology.
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Introduction
Quantum metrology exploits particle entanglement in the probe state to enhance the precision of parameter estimation beyond what is reachable with classical resources (see refs 1,2 for reviews). The role of particle entanglement in the estimation of a single parameter has been clarified^{3,4,5,6} and investigated experimentally in MachZehnder interferometers (MZIs)^{7}. However, much less is known about the role of particle entanglement in the joint estimation of multiple parameters. Multiparameter estimation is relevant in many practical applications, including quantum imaging^{8}, quantum process tomography^{9}, as well as probing of biological samples^{10}. Interestingly, the theory of multiphase estimation does not follow trivially from what is known about the single parameter case^{11,12}. Indeed, ultimate multiphase estimation bounds are not saturable in general^{13}, due to the noncommutativity of the operators generating the phase shift transformations^{14,15}. First insights on this scenario have been recently reported^{16,17,18,19,20,21,22}.
A natural platform for multiparameter quantum metrology is provided by multiport interferometry, generalizing conventional twomode interferometry. Recent progresses in the realization of multiport devices have been achieved by exploiting integrated photonics^{23,24,25,26,27,28,29,30,31}. Three and fourport beamsplitters (tritters and quarters) have been produced with integrated optics^{31,32,33,34}. This paves the way toward the realization of multiarm interferometers created by two tritters (quarters) in succession^{35}. Quantumenhanced single parameter estimation in integrated interferometers has been theoretically predicted^{17}, while multiparameter estimation in multiarm interferometers has been examined and compared with the sensitivity achievable by multiple singleparameter estimation^{18}.
In this manuscript we provide conditions of useful particle entanglement for the simultaneous estimation of multiple phases. We study a general multimode scenario where each particle is treated as a qudit. Furthermore, we adapt the theory to the case of multiarm Mach Zehnder interferometers (MMZIs) considering an experimentally relevant framework, with multiphoton Fock states as probe and photon counting measurement. Our analysis generalizes the case of twinFock MZI which has attracted large experimental^{7,36,37,38} and theoretical^{39,40,41} interest for quantumenhanced single phase estimation. From the analysis of the Fisher information and employing an adaptive multiphase estimation, we predict a multiparameter estimation sensitivity beyond the limit achievable with separable qudit probe states.
Results
Multiparameter estimation
We consider here the estimation of a ndimensional vector parameter λ = (λ_{1}, , λ_{n})^{11,12}. In our benchmark, every parameter corresponds to a phase to be estimated in a multiarm interferometer. A general approach (see Fig. 1a) consists in preparing a probe state , applying a λ dependent unitary transformation and performing independent measurements on ν identical copies of the output state . The measurement is described by a positiveoperator valued measure (POVM), i.e. a set of positive operators satisfying , being the probability of the detection event x. Finally, the sequence x ≡ (x_{1}, , x_{ν}) of ν measurement results is mapped into a vector parameter Λ( x ) = (Λ_{1}( x ), , Λ_{n}( x )), representing our estimate of λ . A figure of merit of multiparameter estimation is the covariance matrix
where and is the mean value of the estimator vector. For locally unbiased estimators (i.e. ) the covariance matrix is bounded, via the CramerRao theorem^{11}, as
(in the sense of matrix inequality), where
is the Fisher information matrix (FIM). Notice that Eq. (2) can be derived only when the FIM is invertible. The equality sign in Eq. (2) is saturated asymptotically in ν by the maximum likelihood estimator^{11}. Here we quantify the phase sensitivity by the variance of each estimator, (δλ_{j})^{2} ≡ C_{j,j}. We have
where the first inequality is due to (2) and the second follows from a CauchySchwarz inequality (see Supplementary Information). Since 1/(ν F_{j,j}) is the CramerRao bound for single parameter estimation, inequality (4) tells us that sensitivity in the estimation of λ_{j} can be optimized when fixing all the other parameters to known values. We will also consider
The righthand side inequality in Eqs (4) and (5) is saturated if and only if the FIM is diagonal. Furthermore, the FIM is bounded by the quantum Fisher information matrix (QFIM): F ≤ F_{Q} (in the sense of matrix inequality), where
and is the symmetric logarithmic derivative of ρ_{λ} with respect to parameter λ_{j}, defined by 2 ^{10}. In the single parameter case, the QFIM reduces to a single scalar quantity and it is always possible to find a POVM for which F = F_{Q} and δλ = 1/F_{Q} holds^{42,43}. In contrast, in the multiparameter case, it is generally not possible to achieve the CramerRao bound^{13,14,15}.
Sensitivity bounds for quditseparable states
In the following we consider the estimation of n parameters in a system made of d = n + 1 modes (e.g. the number of arms in a MMZI, see below). A single particle occupying the n + 1 modes is generally indicated as a qudit. The notion of qudit generalizes the concept of qubit (a twomode particle, n = 1) and is relevant in multimode interferometry^{2}. Here we set sensitivity bounds for multiparameter estimation when the probe state is quditseparable. A state of N qudits is said to be quditseparable if it can be written as , where (l = 1, , N) is a single qudit state, p_{k} > 0 and ∑_{k} p_{k} = 1. A state which is not quditseparable is quditentangled. We take the generator of each phase shift, (j = 1, …, n labels the parameter), to be local in the qudit, i.e. it can be written as where is an arbitrary operator acting on the lth qudit. In particular, the transformation does not create entanglement among the N qudits. For simplicity, we will take the same operator for each particle. For a generic separable probe state , the inequality
holds for all possible POVMs (see Supplementary Information), where g_{j,max} and g_{j,min} are the maximum and minimum eigenvalue of , respectively. Inequality (7) gives a bound on the diagonal elements of the FIM. It corresponds, via the inequality (δλ_{j})^{2} ≥ 1/ν F_{j,j}, to a bound on the sensitivity reachable with quditseparable states for the estimation of the single parameter λ_{j}, when all other parameters are set to zero. Inequality (7) can be always saturated by optimal states and measurements (see Supplementary Information). For the estimation of a single parameter, the violation of Eq. (7) is a necessary and sufficient condition of useful qudit entanglement^{2,4}: only those quditentangled states that violate Eq. (7) allow to estimate the parameter λ_{j} with a sensitivity overcoming the one reachable with any quditseparable state. Regarding the simultaneous estimation of multiple parameters, we can use Eq. (7) and the chain of inequalities (4) to obtain
Inequality (8) is a bound of sensitivity in the estimation of the single parameter λ_{j} with quditseparable states, when all the parameters are unknown. Summing Eq. (8) over all parameters, we obtain
According to Eqs (8) and (9), for quditseparable states such that the FIM is invertible, we recover – at best – the shot noise scaling of phase sensitivity, δλ_{j} ∝ N^{−1/2}, which also characterizes single parameter estimation^{3,4}. Notice that the quantity (g_{j,max} − g_{j,min})^{2} is equal to one for any qubit transformation and might be larger than one for general qudit transformations. We finally recall that the phase estimation scenario we are considering – as well as the notion of useful quditentangement – refers to interferometric scheme involving liner qudit transformations and multiple independent measurements done with identical copies of the same probe. Inequalities (8) and (9) have no concern with the quditentanglement of the initial probe state for (nonlinear) parameter dependent processes that entangle/disentangle the probe or nonindependent multiple measurements.
Multimode MachZehnder interferometry
In the following we discuss the estimation of a phase vector ϕ = (ϕ_{1}, …, ϕ_{n}) in a MMZI (see Fig. 1b,c). The MMZI can be obtained by cascading a dmode balanced beamsplitter , a phase shift transformation , being the photonnumber operator for the ith mode, and a second multiport beamsplitter . The dmode beamsplitter is the natural extension of the standard 5050 beamsplitter to more than two optical inputoutput modes^{41}. Hence, the MMZI can be adopted as a benchmark to investigate simultaneous estimation of n = d − 1 optical phases. Indeed, it allows for a direct comparison between classical and quantum probe states and represents a flexible platform for the analysis of multiparameter scenario by changing the unitary transformation of the input and output multiport beamsplitters.
In order to adapt the discussion of the previous section, we consider N particles as input of the MMZI and identify a single particle in the d arms of the interferometer as a qudit, whose Hilbert space has thus dimension d. The generator of phase shift in the jth mode is . One can thus write where as the operator projecting the lth qudit on the jth mode. Finally, g_{j,max} − g_{j,min} = 1 and the inequalities (8) and (9) read
respectively. The violation of one of these inequalities in the MMZI is a signature of useful quditentanglement in the probe state.
The recent experimental implementation of symmetric multiport beamsplitting^{31,32,33,34}, by adopting integrated platforms, paves the way toward the future realization of optical MMZIs. For d = 3 modes, the tritter matrix , corresponding to its unitary transformation , has diagonal elements and offdiagonal elements with i ≠ j. For d = 4 modes, the quarter matrix is and for i ≠ j. The overall matrix for the MMZI is then obtained as . The phase vector is estimated from the measurement of the number of particles in each mode. As probe, we focus on multimode Fock states with a single photon in each input mode of the interferometer^{18}, 1, 1, 1〉 and 1, 1, 1, 1〉 for the three and fourmode MZI, respectively. Here, 1, 1, 1〉 ≡ 1〉_{1} ⊗ 1〉_{2} ⊗ 1〉_{3} (and analogous definition for 1, 1, 1, 1〉), where 1〉_{j} is a Fock state identifying a single particle in the jth mode.
For the threemode MZI, the results of the calculation for F^{−1} are shown in Fig. 2a–c. Analytic expression of the FIM is reported in the Supplementary Information. We observe that Tr[F^{−1}] and the diagonal elements [F^{−1}]_{1,1} and [F^{−1}]_{2,2} depend on the phases ϕ_{1} and ϕ_{2}. Notably, the inequalities (10) are violated at certain optimal values of the parameters, signaling that the Fock state 1, 1, 1〉 contains useful qudit entanglement: we find (see Fig. 2a) and (see Fig. 2b,c), which are smaller than the bound for quditseparable states Tr[F^{−1}] = 0.667 and [F^{−1}]_{j,j} = 0.33 (here N = 3 and n = 2), respectively. Additionally, we observe characteristic features. (i) F ≠ F_{Q}, in particular, the minimum value of Tr[F^{−1}] is greater than the corresponding minimum value of the QFIM: (see Fig. 2a). (ii) The FIM is not always invertible: at the phase values for which det F = 0 the bound (2) is not defined. Around these points (white regions in Fig. 2a–c) [F^{−1}]_{1,1} and/or [F^{−1}]_{2,2} diverge. (iii) The working points to obtain the minimum of the multiparameter bound do not lead to symmetric errors on the single parameters ϕ_{1} and ϕ_{2}. More specifically, when Tr[F^{−1}] = 0.59, the bounds for the error on the single parameters are different: . This is obtained for instance for working point Q_{1} = (ϕ_{1},ϕ_{2}) = (0.892, 2.190), leading to and for working point Q_{2} = (ϕ_{1},ϕ_{2}) = (2.190, 0.892), leading to , see Fig. 2a. In summary, with this choice of probe state and measurement it is not possible to saturate the quantum CramerRao inequality simultaneously for the two parameters. Furthermore, according to point (iii) an adaptive estimation strategy (which we discuss below) is necessary to obtain the minimum sensitivity on both parameters with symmetric errors, and thus saturate the multiparameter CramerRao bound.
We have repeated the above analysis for the fourmode interferometer (d = 4) with two unknown phases, ϕ_{1} and ϕ_{2}, and a known control phase ϕ_{0} (see Fig. 1c). This configuration allows a comparison between three and fourarm interferometers for the two parameter estimation. In the latter case the control phase ϕ_{0} gives us an additional degree of freedom. We choose as input the Fock State 1, 1, 1, 1〉. In Fig. 2d–f the results of our calculations are reported for a fixed value of ϕ_{0}, as well as the numerical analysis of det F. We observe that as in the previous cases the FIM depends on the value of the parameter to be estimated. Furthermore, also in the fourmode the achievable sensitivity falls below the bound (10) for separable states: we have , and which are below the bounds 0.5 and 0.25 given by Eq. (10) (N = 4 and n = 2, here), respectively. The most notable difference with respect to the previous case is that the QCRB is achieved, for instance in working point O_{1} = [π, π]. In addition, both diagonal terms are equivalent and only a two step adaptive protocol is needed to reach the QCRB for any arbitrary phase vector (see discussion below).
We have also compared the obtained results with the one achievable with other probe states. For instance, we consider a set of distinguishable particles (where q〉 stands for a single photon on mode k_{q}), or an input coherent state on input mode k_{1} with for d = 3 (α = 2 for d = 4) and no phase reference. We obtain for both and , within the bound Tr[F^{−1}] ≥ 0.667 given by Eq. (10) for separable inputs. Similarly, for both and , within the bound Tr[F^{−1}] ≥ 0.5. Results are summarized in Tables A and B.
Adaptive phase estimation
In this section we present the adaptive estimation protocols required to maximize the precision on the simultaneous estimation of two arbitrary phases in a three and four mode MZI. The resources (the number of independent measurements ν) are split between multiple steps. A first step is needed to obtain a rough estimate of the unknown phases and requires a small subset of the resources which becomes negligible when the number of repetitions ν of the experiment is large enough. The subsequent steps exploit the available information to optimize the estimation procedure.
Regarding the threemode interferometer, the above analysis has identified working points (Q_{1} and Q_{2}) where the minimum uncertainty for the estimation of the two phases ϕ_{1} and ϕ_{2} does not give the same error on the two individual parameters. To overcome this limitation – and obtain approximatively a symmetric error in the joint estimation of the two phases – we exploited a threestep adaptive algorithm. The protocol requires ν independent measurements and the adoption of controlled phase shifts ψ_{i} on modes k_{i}, with i = 1, 2, which have to be tuned during the protocol to perform the estimation at different working points (see in Fig. 1b). In a first step, we set ψ_{1,2} = 0 and obtain a rough estimate of the phases ϕ_{i} after a number of measurements much smaller than ν. Then, in step 2 the tunable phases ψ_{i} are adjusted so that ϕ_{i} + ψ_{i} on arms 1 and 2 are set to be close to the working point Q_{1}. In this step essentially half of the remaining resources are spent so as to obtain and with an adequate estimator. Here , represent respectively the estimation and the uncertainty of ϕ_{i} around working point Q_{1}. In step 3 the same procedure is repeated for working point Q_{2}. Finally the tunable phases ψ_{1.2} are subtracted so to recover ϕ_{1,2} ± δϕ_{1,2}. The results of the adaptive algorithm are shown in Fig. 3a–d. Half of the measurements (ν_{1} = ν/2) are performed at point Q_{1}, where and , while the other half (ν_{2} = ν/2) are performed at point Q_{2}, where and . The expected error on a single phase δϕ_{i} after the two steps is then obtained as an appropriate combination of the values on the points Q_{i}. More specifically, as the Fisher information is additive, the overall FIM reads F = ν_{1}F_{1} + ν_{2}F_{2}, where F_{i} is the FIM in working points Q_{i}. We observe that the protocol permits to achieve the bound of the working point, which for ν_{1} = ν_{2} is . Note that the bound is lower than the bound (10) for separable states .
The adaptive scheme for the fourmode interferometer is slightly different: in this case there are optimal working points, as the point O_{1}, see Fig. 2, where QCRB is achieved for both phases. To reach the QCRB for arbitrary phases, we thus apply a twostep adaptive protocol. In the first step, we obtain a rough estimate of the parameters with an initial error δ. Then, in the second step we apply two supplementary phases ψ_{1} and ψ_{2} to translate the working point of the protocol to the neighbourhood of O_{1}. It should be noticed that a convergent estimation protocol in the second step requires to set ϕ_{0} such that the quantity Tr[F^{−1}] has no singularities. Note that the more ϕ_{0} deviates from ϕ_{0} = 0, the larger is the regular region around O_{1} (see Supplementary Information). The price to pay is a slightly increasing the error in the estimation process. The value of ϕ_{0} has to be chosen in order to move the singularity away from a neighbourhood of O_{1} larger than the inital error δ of the first step. The results of the protocol for the fourmode case with ϕ_{0} = 0.01 are then shown in Fig. 4a,b. Similarly to the threemode case, we observe that the protocol permits to achieve the bound of the working point, which is for ϕ_{0} = 0.01 (plane in Fig. 4), while the quantum CramerRao bound reads . This shows that achieving a convergent numerical protocol leads to a slight decrease in phase sensitivity due to singular points in the neighborhood of the working regions. Also in this case, the adaptive protocol allows to reach a sensitivity overcoming the bound of separable state for any vector parameter.
Conclusions
In this manuscript we have developed the general theory of quantumenhanced multiphase estimation. In particular, we provide conditions of useful quditentanglement for the simultaneous estimation of multiple phases below the ultimate sensitivity limit achievable with quditseparable states. We have focused on interferometers involving linear qudit transformations and multiple independent measurements. In a realistic experimental scenario, using multimode MachZehnder interferometers and photocounting measurements, Fock state probes can be exploited for multiphase estimation with quantumenhancement phase sensitivity. With respect to the estimation of a single phase, where Fock states are known to be a useful resource, our analysis evidences a rich scenario: most notably, the phase sensitivity strongly depends on the phase value (the CramerRao bound being not always definite) and on the interferometer configurations such as the three and fourmode interferometers. Finally, we discuss and numerically simulate an adaptive estimation protocol which permits to achieve the expected bounds for any vector parameter. The adaptive strategy becomes crucial in multiparameter estimation since the simultaneous saturation of the ultimate limits for all parameters is in general not guaranteed.
During the completion of this manuscript, a first implementation of a tritterbased interferometer for singlephase estimation has been reported^{45}.
Additional Information
How to cite this article: Ciampini, M. A. et al. Quantumenhanced multiparameter estimation in multiarm interferometers. Sci. Rep. 6, 28881; doi: 10.1038/srep28881 (2016).
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Acknowledgements
This work was supported by ERCStarting Grant 3DQUEST (3DQuantum Integrated Optical Simulation; grant agreement no. 307783, http://www.3dquest.eu), EUSTREP Project QIBEC and PRIN project Advanced Quantum Simulation and Metrology (AQUASIM). LP acknowledges financial support by MIUR through FIRB Project No. RBFR08H058.
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M.A.C., N.S., C.V., L.P., A.S. and F.S. contributed to design the ideas, perform the calculations, analyse the results and write the manuscript.
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Ciampini, M., Spagnolo, N., Vitelli, C. et al. Quantumenhanced multiparameter estimation in multiarm interferometers. Sci Rep 6, 28881 (2016). https://doi.org/10.1038/srep28881
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