SUFFICIENCY IN QUANTUM STATISTICAL INFERENCE: A SURVEY WITH EXAMPLES

This paper attempts to give an overview about sufficiency in the setting of quantum statistics. The basic concepts are treated in parallel to the measure theoretic case. It turns out that several classical examples and results have a noncommutative analogue. Some of the results are presented without proof (but with exact references) and the presentation is intended to be self-contained. The main examples discussed in the paper are related to the Weyl algebra and to the exponential family of states. The characterization of sufficiency in terms of quantum Fisher information is a new result.

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Taming singularities of the quantum Fisher information

International Journal of Quantum Information ◽

10.1142/s0219749921400049 ◽

2021 ◽

Author(s):

Aaron Z. Goldberg ◽

José L. Romero ◽

Ángel S. Sanz ◽

Luis L. Sánchez-Soto

Keyword(s):

Density Matrix ◽

Fisher Information ◽

Statistical Models ◽

Probability Distributions ◽

Quantum Fisher Information ◽

Quantum Metrology ◽

Quantum Statistical ◽

Information Matrices ◽

Parameter Dependent ◽

Ultimate Limit

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.

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On the discontinuity of the quantum Fisher information for quantum statistical models with parameter dependent rank

Journal of Physics A Mathematical and Theoretical ◽

10.1088/1751-8121/ab599b ◽

2019 ◽

Vol 53 (2) ◽

pp. 02LT01 ◽

Cited By ~ 7

Author(s):

Luigi Seveso ◽

Francesco Albarelli ◽

Marco G Genoni ◽

Matteo G A Paris

Keyword(s):

Fisher Information ◽

Statistical Models ◽

Quantum Fisher Information ◽

Quantum Statistical ◽

Parameter Dependent

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About the quantum Fisher information of nearly pure quantum statistical models

International Journal of Quantum Information ◽

10.1142/s0219749919410223 ◽

2020 ◽

Vol 18 (01) ◽

pp. 1941022

Author(s):

Matteo G. A. Paris

Keyword(s):

Fisher Information ◽

Statistical Models ◽

Logarithmic Derivative ◽

Quantum Fisher Information ◽

Independent State ◽

Approximate Analytic Expression ◽

Weak Source ◽

Pure States ◽

Analytic Expression ◽

Quantum Statistical

We address nearly pure quantum statistical models, i.e. situations where the information about a parameter is encoded in pure states weakly perturbed by the mixing with a parameter independent state, mimicking a weak source of noise. We show that the symmetric logarithmic derivative is left unchanged, and find an approximate analytic expression for the quantum Fisher information (QFI) which provides bounds on how much a weak source of noise may degrade the QFI.

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