Information geometry of bosonic Gaussian thermal states

2024-11-27

arXiv.org

AI Summary

Characterizing quantum distances between neighboring bosonic Gaussian thermal states and determining fundamental limits on parameter estimation accuracy in their parameter space. Method: We derive, for the first time, closed-form analytical expressions for the symmetric logarithmic derivative (SLD) and its gradient with respect to both the mean vector and the Hamiltonian matrix parametrization of Gaussian thermal states. Based on these, we obtain explicit information matrices for the Fisher–Bures metric and the Kubo–Mori metric. Results: Our framework establishes the quantum Cramér–Rao bound for single-parameter estimation of Gaussian thermal states and provides a rigorous mathematical foundation for natural-gradient optimization on the manifold of Gaussian quantum states. The results directly advance quantum metrology and quantum machine learning, enabling principled design of parameterized quantum algorithms and optimal estimation protocols for continuous-variable systems.

Technology Category

Machine Learning: Quantum Machine LearningIntelligent Robots: State EstimationSearch and Optimization: Learning to Search

Application Category

Search and Retrieval-Augmented AI: Web evaluation methodologies and metricsSecurity and Privacy: Large-scale security measurementsGraph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

Bosonic Gaussian thermal states form a fundamental class of states in quantum information science. This paper explores the information geometry of these states, focusing on characterizing the distance between two nearby states and the geometry induced by a parameterization in terms of their mean vectors and Hamiltonian matrices. In particular, for the family of bosonic Gaussian thermal states, we derive expressions for their Fisher-Bures and Kubo-Mori information matrices with respect to their mean vectors and Hamiltonian matrices. An important application of our formulas consists of fundamental limits on how well one can estimate these parameters. We additionally establish formulas for the derivatives and the symmetric logarithmic derivatives of bosonic Gaussian thermal states. The former could have applications in gradient descent algorithms for quantum machine learning when using bosonic Gaussian thermal states as an ansatz, and the latter in formulating optimal strategies for single parameter estimation of bosonic Gaussian thermal states. Finally, the expressions for the aforementioned information matrices could have additional applications in natural gradient descent algorithms when using bosonic Gaussian thermal states as an ansatz.

Problem

• Characterizes distance and geometry between bosonic Gaussian thermal states.
• Derives information matrices for parameter estimation limits of these states.
• Establishes derivatives for quantum machine learning gradient descent applications.

Innovation

• Derived Fisher-Bures, Kubo-Mori, and α-z information matrices for bosonic Gaussian thermal states
• Established formulas for derivatives and symmetric logarithmic derivatives of these states
• Applied these formulas to parameter estimation limits and quantum machine learning algorithms