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arXiv:2108.05976 · International Journal of Quantum Information

Taming singularities of the quantum Fisher information

Aaron Z. Goldberg, José L. Romero, Ángel S. Sanz, Luis Lorenzo Sánchez-Soto

Published Nov 17, 2021• 18 citations
arXivHTMLPDF

Abstract

Quantum Fisher information matrices (QFIMs) are fundamental to estimation theory: they encode the ultimate limit for the sensitivity with which a set of parameters can be estimated using a given probe. Since the limit invokes the inverse of a QFIM, an immediate question is what to do with singular QFIMs. Moreover, the QFIM may be discontinuous, forcing one away from the paradigm of regular statistical models. These questions of nonregular quantum statistical models are present in both single- and multiparameter estimation. Geometrically, singular QFIMs occur when the curvature of the metric vanishes in one or more directions in the space of probability distributions, while QFIMs have discontinuities when the density matrix has parameter-dependent rank. We present a nuanced discussion of how to deal with each of these scenarios, stressing the physical implications of singular QFIMs and the ensuing ramifications for quantum metrology.

DOI
10.1142/s0219749921400049

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Quantum Fisher information matrices (QFIMs) are fundamental to estimation theory: they encode the ultimate limit for the sensitivity with which a set of parameters can be estimated using a given probe. Since the limit invokes the inverse of a QFIM, an immediate question is what to do with singular QFIMs. Moreover, the QFIM may be discontinuous, forcing one away from the paradigm of regular statistical models. These questions of nonregular quantum statistical models are present in both single- and multiparameter estimation. Geometrically, singular QFIMs occur when the curvature of the metric vanishes in one or more directions in the space of probability distributions, while QFIMs have discontinuities when the density matrix has parameter-dependent rank. We present a nuanced discussion of how to deal with each of these scenarios, stressing the physical implications of singular QFIMs and the ensuing ramifications for quantum metrology.
Abstract We address the discontinuities of the quantum Fisher information (QFI) that may arise when the parameter of interest takes values that change the rank of the quantum statistical model. We revisit the classical and the quantum Cramér–Rao theorems, show that they do not hold in these limiting cases, and discuss how this impacts on the relationship between the QFI and the Bures metric. In order to illustrate the metrological implications of our findings, we present two paradigmatic examples, where we discuss in detail the role of the discontinuities. We show that the usual equivalence between the variance of the maximum likelihood estimator and the inverse of the QFI breaks down.
We show that two quantities in quantum metrology that were thought to be the same, the quantum Fisher information matrix and the Bures metric, are not the same. They differ at points at which the rank of the density matrix changes. The quantum Fisher information matrix is discontinuous at these points. However, these discontinuities are removable in some sense. We show that the expression given by the Bures metric represents the continuous version of the quantum Fisher information matrix. We also derive an explicit formula for the Bures metric for both singular and nonsingular density matrices.
Abstract Quantum Fisher information matrix (QFIM) is a core concept in theoretical quantum metrology due to the significant importance of quantum Cramér–Rao bound in quantum parameter estimation. However, studies in recent years have revealed wide connections between QFIM and other aspects of quantum mechanics, including quantum thermodynamics, quantum phase transition, entanglement witness, quantum speed limit and non-Markovianity. These connections indicate that QFIM is more than a concept in quantum metrology, but rather a fundamental quantity in quantum mechanics. In this paper, we summarize the properties and existing calculation techniques of QFIM for various cases, and review the development of QFIM in some aspects of quantum mechanics apart from quantum metrology. On the other hand, as the main application of QFIM, the second part of this paper reviews the quantum multiparameter Cramér–Rao bound, its attainability condition and the associated optimal measurements. Moreover, recent developments in a few typical scenarios of quantum multiparameter estimation and the quantum advantages are also thoroughly discussed in this part.
For a given quantum state used in sensing, the quantum Cramér-Rao bound (QCRB) sets a fundamental limit on the precision achievable by an unbiased estimator of an unknown parameter, determined by the inverse of the quantum Fisher information (QFI). The QFI serves as an upper bound on the classical Fisher information (CFI), representing the maximum extractable information about the unknown parameter from measurements on a physical system. Thus, a central goal in quantum parameter estimation is to find a measurement, described by a POVM, that saturates the QFI (achieves maximum CFI), and thereby achieves the QCRB. In the idealization that one uses pure states and unitary encodings for sensing, discontinuities can appear in the CFI but not the QFI. In this article, we demonstrate that these discontinuities are important features, quantifying how much Fisher information is lost in the presence of noise. We refer to this as the Fisher information "fragility". We present a simple framework for understanding how discontinuities increase fragility through Jensen's inequality, and demonstrate how one can use this framework to design more robust POVMs for quantum advantage in metrology.
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.
Abstract Multiparameter quantum estimation theory aims to determine simultaneously the ultimate precision of all parameters contained in the state of a given quantum system. Determining this ultimate precision depends on the quantum Fisher information matrix (QFIM) which is essential to obtaining the quantum Cramér–Rao bound. This is the main motivation of this work which concerns the computation of the analytical expression of the QFIM. Inspired by the results reported in J. Phys. A: Math. Theor. 52 035304 (2019), the general formalism of the multiparameter quantum estimation theory of quantum Gaussian states in terms of their first and second moments is given. We give the analytical formulas of right logarithmic derivative and symmetric logarithmic derivative operators. Then we derive the general expressions of the corresponding quantum Fisher information matrices. We also derive an explicit expression of the condition which ensures the saturation of the quantum Cramér–Rao bound in estimating several parameters. Finally, we examine some examples to clarify the use of our results.