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Quantum Fisher information matrices from Rényi relative entropies

Mark M. Wilde

TL;DR

This work develops a unified framework for quantum generalizations of the Fisher information, derived as Hessians of smooth quantum divergences built from Rényi-type entropies. It proves that the log-Euclidean Rényi information matrix equals the Kubo–Mori information matrix for all admissible $\alpha$, and that the geometric Rényi information matrix equals the right-logarithmic derivative (RLD) information matrix (also for Belavkin–Staszewski). It then derives a general α-z information matrix I_{α,z}(θ) with explicit eigenbasis form, showing its convergence to KM as $\alpha\to1$ or $z\to\infty$, and specializes to Petz–Rényi and sandwiched Rényi cases with integral representations and monotonicity properties. The paper further provides concrete formulas for parameterized thermal and time-evolved states, including Quantum Boltzmann machines, and reveals a cq-state decomposition that yields convexity of these information matrices. The results offer multiple convergent geometries for quantum estimation and learning, with potential impact on quantum thermodynamics, quantum machine learning, and quantum information geometry.

Abstract

Quantum generalizations of the Fisher information are important in quantum information science, with applications in high energy and condensed matter physics and in quantum estimation theory, machine learning, and optimization. One can derive a quantum generalization of the Fisher information matrix in a natural way as the Hessian matrix arising in a Taylor expansion of a smooth divergence. Such an approach is appealing for quantum information theorists, given the ubiquity of divergences in quantum information theory. In contrast to the classical case, there is not a unique quantum generalization of the Fisher information matrix, similar to how there is not a unique quantum generalization of the relative entropy or the Rényi relative entropy. In this paper, I derive information matrices arising from the log-Euclidean, $α$-$z$, and geometric Rényi relative entropies, with the main technical tool for doing so being the method of divided differences for calculating matrix derivatives. Interestingly, for all non-negative values of the Rényi parameter $α$, the log-Euclidean Rényi relative entropy leads to the Kubo-Mori information matrix, and the geometric Rényi relative entropy leads to the right-logarithmic derivative Fisher information matrix. Thus, the resulting information matrices obey the data-processing inequality for all non-negative values of the Rényi parameter $α$ even though the original quantities do not. Additionally, I derive and establish basic properties of $α$-$z$ information matrices resulting from the $α$-$z$ Rényi relative entropies. For parameterized thermal states and time-evolved states, I establish formulas for their $α$-$z$ information matrices and hybrid quantum-classical algorithms for estimating them, with applications in quantum Boltzmann machine learning.

Quantum Fisher information matrices from Rényi relative entropies

TL;DR

This work develops a unified framework for quantum generalizations of the Fisher information, derived as Hessians of smooth quantum divergences built from Rényi-type entropies. It proves that the log-Euclidean Rényi information matrix equals the Kubo–Mori information matrix for all admissible , and that the geometric Rényi information matrix equals the right-logarithmic derivative (RLD) information matrix (also for Belavkin–Staszewski). It then derives a general α-z information matrix I_{α,z}(θ) with explicit eigenbasis form, showing its convergence to KM as or , and specializes to Petz–Rényi and sandwiched Rényi cases with integral representations and monotonicity properties. The paper further provides concrete formulas for parameterized thermal and time-evolved states, including Quantum Boltzmann machines, and reveals a cq-state decomposition that yields convexity of these information matrices. The results offer multiple convergent geometries for quantum estimation and learning, with potential impact on quantum thermodynamics, quantum machine learning, and quantum information geometry.

Abstract

Quantum generalizations of the Fisher information are important in quantum information science, with applications in high energy and condensed matter physics and in quantum estimation theory, machine learning, and optimization. One can derive a quantum generalization of the Fisher information matrix in a natural way as the Hessian matrix arising in a Taylor expansion of a smooth divergence. Such an approach is appealing for quantum information theorists, given the ubiquity of divergences in quantum information theory. In contrast to the classical case, there is not a unique quantum generalization of the Fisher information matrix, similar to how there is not a unique quantum generalization of the relative entropy or the Rényi relative entropy. In this paper, I derive information matrices arising from the log-Euclidean, -, and geometric Rényi relative entropies, with the main technical tool for doing so being the method of divided differences for calculating matrix derivatives. Interestingly, for all non-negative values of the Rényi parameter , the log-Euclidean Rényi relative entropy leads to the Kubo-Mori information matrix, and the geometric Rényi relative entropy leads to the right-logarithmic derivative Fisher information matrix. Thus, the resulting information matrices obey the data-processing inequality for all non-negative values of the Rényi parameter even though the original quantities do not. Additionally, I derive and establish basic properties of - information matrices resulting from the - Rényi relative entropies. For parameterized thermal states and time-evolved states, I establish formulas for their - information matrices and hybrid quantum-classical algorithms for estimating them, with applications in quantum Boltzmann machine learning.

Paper Structure

This paper contains 34 sections, 39 theorems, 329 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Summary of results
  4. Paper organization
  5. Preliminaries
  6. Classical Fisher information
  7. Quantum generalizations of Fisher information
  8. Kubo--Mori information matrix from log-Euclidean Rényi relative entropies
  9. RLD Fisher information matrix from geometric relative entropies
  10. RLD Fisher information matrix from geometric Rényi relative entropies
  11. RLD Fisher information matrix from Belavkin--Staszewski relative entropy
  12. $\alpha$-$z$ Information matrices from $\alpha$-$z$ Rényi relative entropies
  13. Special cases of $\alpha$-$z$ information matrices
  14. Kubo--Mori information matrix
  15. Petz--Rényi information matrices
  16. ...and 19 more sections

Key Result

Theorem 1

Let $\left(\rho(\theta)\right)_{\theta\in\Theta}$ be a second-order differentiable parameterized family of positive definite states, where $\Theta\subseteq\mathbb{R}^{L}$ is open and $L\in\mathbb{N}$. For every smooth quantum divergence $\boldsymbol{D}$, there exists a function $\zeta\colon\mathbb{R where the quantum information matrix $I_{\zeta}(\theta)$ is defined in terms of its elements as $\

Figures (2)

  • Figure 1: Contour $\gamma_{R}^{+}$ for $t>0$.
  • Figure 2: Contour $\gamma_{R}^{-}$ for $t<0$.

Theorems & Definitions (77)

  • Theorem 1: Morozova1991Petz1996
  • Definition 2: Kubo--Mori information matrix
  • Theorem 3
  • proof
  • Remark 4
  • Theorem 5
  • proof
  • Remark 6
  • Theorem 7
  • proof
  • ...and 67 more