Geometry of Quantum Statistical Inference (original) (raw)

Statistical geometry in quantum mechanics

Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 1998

A statistical model M is a family of probability distributions, characterised by a set of continuous parameters known as the parameter space. This possesses natural geometrical properties induced by the embedding of the family of probability distributions into the space of all square-integrable functions. More precisely, by consideration of the square-root density function we can regard M as a submanifold of the unit sphere S in a real Hilbert space H. Therefore, H embodies the 'state space' of the probability distributions, and the geometry of the given statistical model can be described in terms of the embedding of M in S. The geometry in question is characterised by a natural Riemannian metric (the Fisher-Rao metric), thus allowing us to formulate the principles of classical statistical inference in a natural geometric setting. In particular, we focus attention on the variance lower bounds for statistical estimation, and establish generalisations of the classical Cramér-Rao and Bhattacharyya inequalities, described in terms of the geometry of the underlying real Hilbert space. As a comprehensive illustration of the utility of the geometric framework, the statistical model M is then specialised to the case of a submanifold of the state space of a quantum mechanical system. This is pursued by introducing a compatible complex structure on the underlying real Hilbert space, which allows the operations of ordinary quantum mechanics to be reinterpreted in the language of real Hilbert space geometry. The application of generalised variance bounds in the case of quantum statistical estimation leads to a set of higher order corrections to the Heisenberg uncertainty relations for canonically conjugate observables.

Geometric perspective on quantum parameter estimation

AVS Quantum Science

Quantum metrology holds the promise of an early practical application of quantum technologies, in which measurements of physical quantities can be made with much greater precision than what is achievable with classical technologies. In this review, we collect some of the key theoretical results in quantum parameter estimation by presenting the theory for the quantum estimation of a single parameter, multiple parameters, and optical estimation using Gaussian states. We give an overview of results in areas of current research interest, such as Bayesian quantum estimation, noisy quantum metrology, and distributed quantum sensing. We address the question how minimum measurement errors can be achieved using entanglement as well as more general quantum states. This review is presented from a geometric perspective. This has the advantage that it unifies a wide variety of estimation procedures and strategies, thus providing a more intuitive big picture of quantum parameter estimation. CONTENTS I. Introduction 1 II. Classical estimation theory 2 A. Fundamentals of estimation theory 3 B. Expectation values and covariance 3 C. Bounds on the covariance matrix 4 D. The Cramér-Rao bound 5 III. Geometry of estimation theory 6 A. The probability simplex 6 B. The Fisher-Rao metric and statistical distance 6 C. Relative entropy 7 IV. Single parameter quantum estimation 8 A. Quantum model of precision measurements 8 B. The quantum Fisher information 8 C. Distance measures in quantum estimation 10 D. The Symmetric Logarithmic Derivative 10 E. The quantum Cramér-Rao bound 12 F. Biased Estimators 13 G. The role of entanglement 13 H. Non-entangling strategies 15 I. Optimal estimation strategies 15 J. Numerical approaches 16 V. Multi-parameter quantum estimation 16 A. The quantum Fisher information matrix 16 B. The quantum Cramér-Rao bound 18 C. Saturating the quantum Cramér-Rao bound 18 D. Simultaneous versus sequential estimation 19 E. The Right Logarithmic Derivative 21 F. Kubo-Mori information 23 G. Wigner-Yanase skew information 23 H. Bayesian quantum estimation theory 24

Generalised Heisenberg relations for quantum statistical estimation

Physics Letters A, 1997

A geometric framework for quantum statistical estimation is used to establish a series of corrections to the Heisenberg uncertainty relations for canonically conjugate variables. These results apply when the true state of the system belongs to a one-parameter family of unitarily related states, and we are required to estimate the value of the parameter. @ 1997 Elsevier Science B.V.

On Quantum Statistical Inference

Interest in problems of statistical inference connected to measurements of quantum systems has recently increased substantially, in step with dramatic new developments in experimental techniques for studying small quantum systems. Furthermore, theoretical developments in the theory of quantum measurements have brought the basic mathematical framework for the probability calculations much closer to that of classical probability theory. The present paper reviews this field and proposes and interrelates a number of new concepts for an extension of classical statistical inference to the quantum context.

On Quantum Statistical Inference, II

Eprint Arxiv Quant Ph 0307191, 2003

Quantum Statistical Inference

2020

In this paper, inspired by the "Minimum Description Length Principle" in classical Statistics, we introduce a new method for predicting the outcomes of a quantum measurement and for estimating the state of a quantum system with minimum quantum complexity, while, at the same time, avoiding overfitting.

On Quantum Statistical Inference Ole E

2003

On asymptotic quantum statistical inference

We study asymptotically optimal statistical inference concerning the unknown state of N identical quantum systems, using two complementary approaches: a "poor man's approach" based on the van Trees inequality, and a rather more sophisticated approach using the recently developed quantum form of LeCam's theory of Local Asymptotic Normality. * URL: www.math.leidenuniv.nl/∼gill.

Principles of Quantum Inference

Annals of Physics, 1991

A new approach to quantum state determination is developed using data in the form of observed eigenvectors. An exceedingly natural inversion of such data results when the quantum probability rule is recognised as a conditional. The reversal of this conditional via Bayesian methods results in an inferred probability density over states which readily reduces to a density matrix estimator. The inclusion of concepts drawn from communication theory then defines an Optimal State Determination Problem which is explored on Hilbert spaces of arbitrary finite dimensionality.

the Quantum Estimation Problem

2014

Information metrics give lower bounds for the estimation of parameters. The Cencov-Morozova-Petz Theorem classifies the monotone quantum Fisher metrics. The optimum bound for the quantum estimation problem is offered by the metric which is obtained from the symmetric logarithmic derivative. To get a better bound, it means to go outside this family of metrics, and thus inevitably, to relax some general conditions. In the paper we defined logarithmic derivatives through a phase-space correspondence. This introduces a function which quantifies the deviation from the symmetric derivative. Using this function we have proved that there exist POVMs for which the new metric gives a higher bound from that of the symmetric derivative. The analysis was performed for the one qubit case.

Geometry of Quantum Statistical Inference (original) (raw)

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