^{1,a)}

### Abstract

The two principal/immediate influences—which we seek to interrelate here—upon the undertaking of this study are papers of Życzkowski and Słomczyński [J. Phys. A34, 6689 (2001)] and of Petz and Sudár [J. Math. Phys.37, 2262 (1996)]. In the former work, a metric (the Monge one, specifically) over generalized Husimi distributions was employed to define a distance between two arbitrary density matrices. In the Petz-Sudár work (completing a program of Chentsov), the quantum analog of the (classically unique) Fisher information (monotone) metric of a probability simplex was extended to define an uncountable infinitude of Riemannian (also monotone) metrics on the set of positive definite density matrices. We pose here the questions of what is the specific/*unique* Fisher information metric for the (classically defined) Husimi distributions and how does it relate to the *infinitude* of (quantum) metrics over the density matrices of Petz and Sudár? We find a highly proximate (small relative entropy) relationship between the probability distribution (the quantum Jeffreys’ prior) that yields quantum universal data compression, and that which (following Clarke and Barron) gives its classical counterpart. We also investigate the Fisher information metrics corresponding to the *escort* Husimi, positive- and certain Gaussian probability distributions, as well as, in some sense, the discrete Wigner *pseudoprobability*. The *comparative noninformativity* of prior probability distributions—recently studied by Srednicki [Phys. Rev. A71, 052107 (2005)]—formed by normalizing the volume elements of the various information metrics, is also discussed in our context.

The author expresses gratitude to the Kavli Institute for Theoretical Physics (KITP) for computational support in this research and to C. Krattenthaler for deriving the expressions (30) and (33) and to M. Trott for assistance with certain MATHEMATICA computations.

I. INTRODUCTION

II. MONOTONE METRICS

III. COMPARATIVE NONINFORMATIVITIES

A. Bures prior

B. Morozova-Chentsov prior

C. Hilbert-Schmidt prior

IV. UNIVERSAL DATA COMPRESSION

V. ESCORT-HUSIMI DISTRIBUTIONS

A. The case

B. The cases

C. Tangential components

D. Radial components

VI. POSITIVE -REPRESENTATION FOR TLQS

VII. GAUSSIAN DISTRIBUTION

VIII. DISCRETE WIGNER FUNCTION FOR A QUBIT

IX. SCALAR CURVATURE

X. DISCUSSION

XI. SUMMARY

### Key Topics

- Probability theory
- 39.0
- Quantum information
- 23.0
- Entropy
- 22.0
- Qubits
- 10.0
- Monte Carlo methods
- 7.0

## Figures

The monotone function that yields the *tangential* component of the Fisher information metric over the trivariate Husimi probability distributions for the two-level quantum systems.

The monotone function that yields the *tangential* component of the Fisher information metric over the trivariate Husimi probability distributions for the two-level quantum systems.

The radial components of *any* monotone metric and that of the Fisher information metric derived from the family of trivariate Husimi distributions over the TLQS. The one for the (nondenumerably infinite) class dominates that for .

The radial components of *any* monotone metric and that of the Fisher information metric derived from the family of trivariate Husimi distributions over the TLQS. The one for the (nondenumerably infinite) class dominates that for .

Monotonically *decreasing* function obtained by equating the volume element of to that of a generic monotone metric (1).

Monotonically *decreasing* function obtained by equating the volume element of to that of a generic monotone metric (1).

Plots of one-dimensional marginal probability distributions over the radial coordinate of , , , , and . The order of dominance of the curves is . The marginal distributions of and are quite close, as reflected in their small relative entropy .

Plots of one-dimensional marginal probability distributions over the radial coordinate of , , , , and . The order of dominance of the curves is . The marginal distributions of and are quite close, as reflected in their small relative entropy .

The monotone functions , that yield the *tangential* components of the Fisher information metric over the escort-Husimi probability distributions. The steepness of the graphs decreases as increases.

The monotone functions , that yield the *tangential* components of the Fisher information metric over the escort-Husimi probability distributions. The steepness of the graphs decreases as increases.

*Approximation* to the presumed operator monotone function yielding the *tangential* component of for the positive representation over the two-level quantum systems.

*Approximation* to the presumed operator monotone function yielding the *tangential* component of for the positive representation over the two-level quantum systems.

*Approximation* to the *radial* component of for the positive -representation over the two-level quantum systems.

*Approximation* to the *radial* component of for the positive -representation over the two-level quantum systems.

Radial components of and . The latter dominates the former.

Radial components of and . The latter dominates the former.

Scalar curvature of the Fisher information metric for the family of Husimi distributions.

Scalar curvature of the Fisher information metric for the family of Husimi distributions.

Statistical distance as a function of distance from the origin of the Bloch ball—corresponding to the fully mixed state—for any monotone metric, for , and for the Monge (or equivalently, for , Hilbert-Schmidt) metric. The monotone-metric curve dominates that for , which dominates the linear curve for the Monge metric.

Statistical distance as a function of distance from the origin of the Bloch ball—corresponding to the fully mixed state—for any monotone metric, for , and for the Monge (or equivalently, for , Hilbert-Schmidt) metric. The monotone-metric curve dominates that for , which dominates the linear curve for the Monge metric.

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