Quantum Fisher information matrices from Rényi relative entropies

2025-10-02

AI Summary

This work addresses the non-uniqueness of quantum extensions of the Fisher information matrix (QFIM) by systematically deriving QFIMs from three distinct Rényi relative entropies: log-Euclidean, α-z, and geometric Rényi divergences. Using a split-difference method for efficient Hessian computation, we uncover the underlying information-geometric structures: the log-Euclidean divergence yields the Kubo–Mori QFIM, while the geometric Rényi divergence yields the right logarithmic derivative (RLD) QFIM—both satisfying the data processing inequality for all α ≥ 0, thereby extending beyond conventional restrictions. We further derive an analytical expression for the α-z QFIM of parameterized thermal states and design a hybrid quantum-classical estimation algorithm compatible with noisy intermediate-scale quantum (NISQ) devices. This framework provides new theoretical tools and practical methodologies for quantum metrology, thermodynamic characterization of quantum systems, and learning in quantum Boltzmann machines.

Technology Category

• Machine Learning: Quantum Machine Learning
• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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, I establish formulas for their $α$-$z$ information matrices and hybrid quantum-classical algorithms for estimating them, with applications in quantum Boltzmann machine learning.

Problem

• Deriving quantum Fisher information matrices from Rényi relative entropies
• Establishing properties of α-z information matrices for quantum states
• Developing quantum algorithms for estimating information matrices in machine learning

Innovation

• Derives quantum Fisher information matrices from Rényi relative entropies
• Uses divided differences method for calculating matrix derivatives
• Establishes formulas for α-z information matrices in thermal states