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Optica Quantum

Estimation of multiple parameters encoded in the modal structure of light

Alexander Boeschoten, Giacomo Sorelli, Manuel Gessner, Claude Fabre, Nicolas Treps

Published Nov 24, 2025• 1 citation
Publisher

Abstract

We investigate the problem of estimating simultaneously multiple parameters encoded in the shape of the modes on which the light is expanded. For this, we generalize the mode-encoded parameter estimation theory as introduced in Ref. [ Optica 10 , 996 ( 2023 ) 10.1364/OPTICA.491368 ] to a multi-parameter scenario. We derive the general expression for the quantum Fisher information matrix and establish the conditions under which the multi-parameter quantum Cramér-Rao bound is attainable. In specific scenarios, we find that each parameter can be associated with a mode—the detection mode—that is proportional to the derivative of either a single non-vacuum mode or the mean-field mode. For a single non-vacuum mode, the correlation between parameters is determined by the real part of the overlap of these detection modes, while in the case of a strong mean-field by the covariance of the quadrature operators of the derivative modes. In both cases, the attainability of the quantum Cramér-Rao bound is determined by the imaginary part of the overlap of the detection modes. Our findings provide clear criteria for optimal joint estimation of parameters encoded in the modal structure of light, and can be used to benchmark experimental multi-parameter estimations and find optimal estimation strategies by carefully shaping the modes and populating them with non-classical light.

DOI
10.1364/opticaq.568679

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We investigate the problem of estimating simultaneously multiple parameters encoded in the shape of the modes on which the light is expanded. For this, we generalize the mode-encoded parameter estimation theory as introduced in Ref. [ Optica 10 , 996 ( 2023 ) 10.1364/OPTICA.491368 ] to a multi-parameter scenario. We derive the general expression for the quantum Fisher information matrix and establish the conditions under which the multi-parameter quantum Cramér-Rao bound is attainable. In specific scenarios, we find that each parameter can be associated with a mode—the detection mode—that is proportional to the derivative of either a single non-vacuum mode or the mean-field mode. For a single non-vacuum mode, the correlation between parameters is determined by the real part of the overlap of these detection modes, while in the case of a strong mean-field by the covariance of the quadrature operators of the derivative modes. In both cases, the attainability of the quantum Cramér-Rao bound is determined by the imaginary part of the overlap of the detection modes. Our findings provide clear criteria for optimal joint estimation of parameters encoded in the modal structure of light, and can be used to benchmark experimental multi-parameter estimations and find optimal estimation strategies by carefully shaping the modes and populating them with non-classical light.
We investigate the problem of estimating simultaneously multiple parameters encoded in the shape of the modes on which the light is expanded. For this, we generalize the mode-encoded parameter estimation theory as introduced in Ref.[1] to a multi-parameter scenario. We derive the general expression for the Quantum Fisher information matrix and establish the conditions under which the multi-parameter Quantum Cramér-Rao bound is attainable. In specific scenarios, we find that each parameter can be associated with a mode -- the detection mode -- that is proportional to the derivative of either a single non-vacuum mode or the mean-field mode. For a single non-vacuum mode, the correlation between parameters is determined by the real part of the overlap of these detection modes, while in the case of a strong mean-field by the covariance of the quadrature operators of the derivative modes. In both cases, the attainability of the Quantum Cramér-Rao bound is determined by the imaginary part of the overlap of the detection modes. Our findings provide clear criteria for optimal joint estimation of parameters encoded in the modal structure of light, and can be used to benchmark experimental multi-parameter estimations and find optimal measurement strategies by carefully shaping the modes and populating them with non-classical light.
We investigate the ultimate precision achievable in Gaussian quantum metrology. We derive general analytical expressions for the quantum Fisher information matrix and for the measurement compatibility condition, ensuring asymptotic saturability of the quantum Cram\'er-Rao bound, for the estimation of multiple parameters encoded in multimode Gaussian states. We then apply our results to the joint estimation of a phase shift and two parameters characterizing Gaussian phase covariant noise in optical interferometry. In such a scheme, we show that two-mode displaced squeezed input probes with optimally tuned squeezing and displacement fulfill the measurement compatibility condition and enable the simultaneous estimation of all three parameters, with an advantage over individual estimation schemes that quickly rises with increasing mean energy of the probes.
As we enter the era of quantum technologies, quantum estimation theory provides an operationally motivating framework for determining high precision devices in modern technological applications. The aim of any estimation process is to extract information from an unknown parameter embedded in a physical system such as the estimation converges to the true value of the parameter. According to the Cramér-Rao inequality in mathematical statistics, the Fisher information in the case of single-parameter estimation procedures, and the Fisher information matrix in the case of multi-parameter estimation, are the key quantities representing the ultimate precision of the parameters specifying a given statistical model. In quantum estimation strategies, it is usually difficult to derive the analytical expressions of such quantities in a given quantum state. This review provides comprehensive techniques on the analytical calculation of the quantum Fisher information as well as the quantum Fisher information matrix in various scenarios and via several methods. Furthermore, it provides a mathematical transition from classical to quantum estimation theory applied to many freedom quantum systems. To clarify these results, we examine these developments using some examples. Other challenges, including their links to quantum correlations and saturating the quantum Cramér-Rao bound, are also addressed.
Abstract We derive several expressions for the quantum Fisher information matrix (QFIM) for the multi-parameter estimation of multi-mode Gaussian quantum states, the corresponding symmetric logarithmic derivatives, and conditions for saturability of the quantum Cramér–Rao bound. This bound determines the ultimate precision with which parameters encoded into quantum states can be estimated. We include expressions for mixed states, for the case when the Williamson decomposition of the covariance matrix is known, expressions in terms of infinite series, and expressions for pure states. We also discuss problematic behavior when some modes are pure, and present a method that allows the use of expressions that are defined only for mixed states, to compute QFIM for states with any number of pure modes.
Multimode Gaussian quantum light, which includes multimode squeezed and multipartite quadrature entangled light, is a very general and powerful quantum resource with promising applications in quantum information processing and metrology. In this paper, we determine the ultimate sensitivity in the estimation of any parameter when the information about this parameter is encoded in such light, irrespective of the information extraction protocol used in the estimation and of the quantity measured. In addition we show that an appropriate homodyne detection scheme allows us to reach the Quantum Cramer-Rao bound. We show that, for a given set of available quantum resources, the most economical way to maximize the sensitivity is to put the most squeezed state available in a well-defined light mode. This implies that it is not relevant to take advantage of the existence of squeezed fluctuations in other modes, nor of quantum correlations and entanglement between different modes. We finally apply these considerations to the problem of optimal phase shift estimation.
The quantum Cramer-Rao bound places a limit on the mean square error of a parameter estimation procedure, and its numerical value is determined by the quantum Fisher information. For single parameters, this leads to the well- known Heisenberg limit that surpasses the classical shot-noise limit. When estimating multiple parameters, the situation is more complicated and the quantum Cramer-Rao bound is generally not attainable. In such cases, the use of entanglement typically still offers an enhancement in precision. Here, we demonstrate that entanglement is detrimental when estimating some nuisance parameters. In general, we find that the estimation of coupled parameters does not benefit from either classical or quantum correlations. We illustrate this effect in a practical application for optical gyroscopes.