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arXiv:2407.02605 · Physical Review A

Exact quantum Fisher matrix results for distributed phases using multiphoton polarization Greenberger-Horne-Zeilinger states

Jiaxuan Wang, G. S. Agarwal

Published Jan 8, 2025• 8 citations
arXivHTMLPDF

Abstract

Recently, distributed sensing has been extensively studied using squeezed states. While this is an excellent development, it is desirable to investigate the use of other quantum probes, such as entangled states of light. In this paper, we focus on distributed sensing, i.e., estimating multiple unknown phases at different spatial nodes using multiphoton polarization-entangled Greenberger-Horne-Zeilinger states distributed across different nodes. We utilize tools of quantum metrology and calculate the quantum Fisher information matrix (QFIM). However, the QFIM turns out to be singular, hindering the determination of quantum Cramér-Rao bounds for the parameters of interest. Recent experiments have contended with a weaker form of the Cramér-Rao bound, which does not require the inversion of the QFIM. It is desirable to understand how relevant these weaker bounds are and how closely they approach the exact Cramér-Rao bounds. We thus analyze the reason for this singularity and, by removing a redundant phase, obtain a nonsingular QFIM, allowing us to derive exact quantum Cramér-Rao bounds. Using the nonsingular QFIM, we show that the arithmetic average of the distributed phases is Heisenberg limited. We demonstrate that quantum metrological bounds can be saturated by projective measurements, enabling us to determine the Fisher information matrix, which is also singular. We then show how this singularity can be resolved.

DOI
10.1103/physreva.111.012414

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Recently, distributed sensing has been extensively studied using squeezed states. While this is an excellent development, it is desirable to investigate the use of other quantum probes, such as entangled states of light. In this paper, we focus on distributed sensing, i.e., estimating multiple unknown phases at different spatial nodes using multiphoton polarization-entangled Greenberger-Horne-Zeilinger states distributed across different nodes. We utilize tools of quantum metrology and calculate the quantum Fisher information matrix (QFIM). However, the QFIM turns out to be singular, hindering the determination of quantum Cramér-Rao bounds for the parameters of interest. Recent experiments have contended with a weaker form of the Cramér-Rao bound, which does not require the inversion of the QFIM. It is desirable to understand how relevant these weaker bounds are and how closely they approach the exact Cramér-Rao bounds. We thus analyze the reason for this singularity and, by removing a redundant phase, obtain a nonsingular QFIM, allowing us to derive exact quantum Cramér-Rao bounds. Using the nonsingular QFIM, we show that the arithmetic average of the distributed phases is Heisenberg limited. We demonstrate that quantum metrological bounds can be saturated by projective measurements, enabling us to determine the Fisher information matrix, which is also singular. We then show how this singularity can be resolved.
Breaking the standard quantum limit in the sensing of parameters at different spatial locations, such as in a quantum network, is of great importance. Using the framework of quantum Fisher information, many strategies based on squeezed quantum probes and multipath multiphoton or multiqubit entangled states have been considered. In this context there is always the question of what the simplest measurement is that would saturate the quantum Cramér-Rao bound (QCRB). The simplest quantity to measure would be characteristics of photon flux or population distribution in the case of qubits. Previous studies have shown that the error sensitivity in SU(1,1) interferometry, also known by several other terms such as nonlinear interferometry and time-reversed measurements, does saturate the QCRB for single parameters such as phase, displacement, and loss. In this work we reveal the great utility of generalized SU(1,1) interferometry in distributed sensing. Generalized SU(1,1) interferometry is a combination of SU ( m ) and SU(1,1) elements, where m is the number of nodes in the network. The SU ( m ) element is used to produce distributed entanglement starting from a squeezed photonic or matter probe. We demonstrate how error sensitivity measurement or more precisely the method of moment sensitivity at just one output port can saturate or nearly saturate the QCRB and thus results in Heisenberg sensitivity of network sensing.
In quantum multi-parameter estimation, the precision of estimating unknown parameters is bounded by the Cramér-Rao bound (CRB), defined via the inverse of the Fisher information matrix (FIM). However, in certain scenarios—such as distributed quantum sensing—the FIM becomes non-invertible due to parameter redundancy, which depends on the probe state and measurement. This issue is often handled using a weaker form of the CRB, potentially overestimating the uncertainty and underrepresenting achievable precision. Here, we propose an alternative approach by introducing equality constraints to remove redundancy and define the CRB via the Moore-Penrose pseudoinverse of the FIM. This framework enables systematic treatment of both simultaneous estimation and distributed sensing cases. We demonstrate its utility by reanalyzing several known examples within this unified perspective, highlighting improved interpretability and practical relevance. Our results offer a concrete guideline for addressing non-invertible FIMs and enhancing the precision of quantum multi-parameter estimation in realistic scenarios.
Distributed quantum sensing, which estimates a global parameter across distant nodes, has attracted significant interest for applications such as quantum imaging, sensor networks, and global-scale clock synchronization. N00N states are regarded as one of the optimal quantum resources for quantum metrology, enabling the Heisenberg scaling. Recently, the concept of N00N states has been extended to multimode N00N states for quantum-enhanced multiple-parameter estimation. However, the application of multimode N00N states in distributed quantum sensing remains unexplored. Here, we propose a distributed quantum sensing scheme that achieves the Heisenberg scaling using multimode N00N states. We theoretically show that multimode N00N states can reach the Heisenberg scaling by examining both the Cramér-Rao bound and the quantum Cramér-Rao bound. For experimental demonstration, we employ a four-mode 2002 state to estimate the average of two spatially distributed phases, achieving a 2.74 dB sensitivity enhancement over the standard quantum limit. We believe that utilizing multimode N00N states for distributed quantum sensing offers a promising approach for developing entanglement-enhanced sensor networks.
We establish a quantum Fisher information (QFI) duality for distributed quantum sensor networks with local phase encoding. For any NNN-qubit probe state, where NNN denotes the number of sensors, FQ(w⊤θ)+FQ(v⊤θ)≤NF_Q(\boldsymbol{w}^\top \boldsymbolθ) + F_Q(\boldsymbol{v}^\top \boldsymbolθ) \leq NFQ​(w⊤θ)+FQ​(v⊤θ)≤N for all unit orthogonal sensing directions w\boldsymbol{w}w and v\boldsymbol{v}v, with equality for all equatorial states when N=2N=2N=2 and for Greenberger--Horne--Zeilinger (GHZ) states when N≥2N\geq 2N≥2. Heisenberg-limited precision for direction w\boldsymbol{w}w, FQ(w⊤θ)=NF_Q(\boldsymbol{w}^\top \boldsymbolθ)=NFQ​(w⊤θ)=N, saturates the bound and simultaneously forces zero QFI for all other independent directions. This can be interpreted as the condition for parameter privacy in distributed quantum sensing: attaining Heisenberg-limited precision for the sensing target renders all alternative privacy-intrusive estimations impossible.
Abstract The theoretical foundation of quantum sensing is rooted in the Cramér–Rao formalism, which establishes quantitative precision bounds for a given quantum probe. In many practical scenarios, where more than one parameter is unknown, the multi-parameter Cramér–Rao bound (CRB) applies. Since this is a matrix inequality involving the inverse of the quantum Fisher information matrix (QFIM), the formalism breaks down when the QFIM is singular. In this paper, we examine the physical origins of such singularities, showing that they result from an over-parameterization on the metrological level. This is itself caused by emergent metrological symmetries , whereby the same set of measurement outcomes are obtained for different combinations of system parameters. Although the number of effective parameters is equal to the number of non-zero QFIM eigenvalues, the Cramér–Rao formalism typically does not provide information about the effective parameter encoding. Instead, we demonstrate through a series of concrete examples that Bayesian estimation can provide deep insights. In particular, the metrological symmetries appear in the Bayesian posterior distribution as lines of persistent likelihood running through the space of unknown parameters. These lines are contour lines of the effective parameters which, through suitable parameter transformations, can be estimated and follow their own effective CRBs.
Estimating the quantum Fisher information (QFI) is a central task in quantum science and technology, underpinning the benchmarking of measurement devices and the certification of metrological advantages. Owing to the highly nonlinear dependence of the QFI on the quantum state, achieving efficient estimates that reliably converge to the true value, which is a property known as gap closing, has remained a challenge. Here, we address it experimentally by implementing the recently proposed Krylov shadow tomography (KST) protocol on a versatile photonic platform capable of generating Greenberger-Horne-Zeilinger states of up to six qubits with controllable noise. By comparing KST with leading methods under matched resources, we show that it exhibits the predicted gap-closing behavior and yields a substantially lower mean absolute error. Moreover, we demonstrate that the resulting high-accuracy QFI estimates correctly predict the ultimate precision achievable in a phase estimation experiment, as dictated by the quantum Cramér-Rao bound. These results establish KST as a scalable and practical experimental tool for the efficient characterization and utilization of quantum resources in metrological applications.