Quantum Fisher Information Matrices Derive from Rényi Relative Entropies Via Divided Differences

Generalizations of the Fisher matrix play a crucial role in diverse fields, ranging from high energy physics to machine learning, and researchers continually seek robust methods for their derivation. Mark M. Wilde from Cornell University, along with colleagues, now presents a novel approach to constructing these matrices, utilising Rényi relative entropies and a technique involving matrix derivatives. This work establishes a connection between different Rényi relative entropies and well-known matrices like the Kubo-Mori matrix and the right-logarithmic derivative Fisher matrix, demonstrating that the resulting matrices consistently adhere to the data-processing inequality, even when the original quantities do not. Furthermore, the team derives formulas for these matrices applied to thermal states and proposes hybrid quantum-classical algorithms for their estimation, potentially advancing applications in Boltzmann machine learning.

Quantum Fisher information matrices are crucial in quantum information science, with applications in fields like high energy physics, condensed matter physics, and machine learning. Researchers recently developed a systematic method for generalizing these matrices, building upon the concept of smooth divergences, which measure the difference between probability distributions. The team derived these generalized matrices by analyzing the Hessian matrix, a tool from calculus, that arises when expanding these divergences in a Taylor series.

A significant finding is that, unlike classical information theory, multiple valid generalizations of the Fisher matrix exist. Each generalization stems from a different choice of smooth divergence, including the log-Euclidean, α-Rényi, and geometric Rényi relative entropies. This work provides a framework for selecting the most appropriate generalization for specific quantum information tasks. The researchers demonstrated that these generalized matrices connect to established concepts, such as the Kubo-Mori matrix and the right-logarithmic derivative Fisher matrix, under certain conditions.

Fractional Identities and Detailed Proofs

This document presents a series of rigorous mathematical proofs for specific identities within the realm of fractional calculus or related areas of advanced mathematics. The proofs establish relationships between complex expressions involving fractional powers, differences, and integral or derivative operators, meticulously detailing each step and utilizing intermediate results established in accompanying appendices. The proofs focus on evaluating limits and simplifying complex expressions through algebraic manipulation. One key proof establishes a relationship between a limit and a specific fractional power expression, while another extensively manipulates expressions involving fractional powers and differences. While mathematically sound, the document would benefit from clearer definitions of unconventional notation and a broader context for the identities being proven.

Generalizing Fisher Information With Smooth Divergences

This research establishes a systematic approach to generalizing the Fisher information matrix, a fundamental concept in information science with broad applications in estimation theory, machine learning, and quantum information processing. Researchers derived these matrices using smooth divergences, mathematical tools that quantify the difference between probability distributions, and specifically by examining the Hessian matrix resulting from a Taylor expansion of these divergences. A key achievement lies in demonstrating that, unlike the classical case, multiple valid generalizations of the Fisher matrix exist, each arising from different choices of smooth divergence, namely, the log-Euclidean, α-Rényi, and geometric Rényi relative entropies.

Notably, the study reveals specific connections between these generalized matrices and established concepts, such as the Kubo-Mori matrix and the right-logarithmic derivative Fisher matrix, for certain values of the Rényi parameter. Furthermore, the researchers established formulas for calculating these matrices for parameterized thermal states and developed hybrid classical-quantum algorithms for their estimation, potentially advancing applications in Boltzmann machine learning.