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Information geometry of bosonic Gaussian thermal states

Zixin Huang, Mark M. Wilde

TL;DR

This work develops the information-geometric characterization of bosonic Gaussian thermal states parameterized by their mean vector $\mu$ and Hamiltonian $H$. It derives explicit formulas for the Fisher–Bures, Kubo–Mori, and $\alpha$-$z$ information matrices for these states, along with derivative expressions and symmetric logarithmic derivatives, enabling both multiparameter estimation bounds and single-parameter optimization strategies. The results are illustrated by a squeezed-thermal single-mode example and have potential applications in quantum metrology, gradient-based quantum machine learning, and natural gradient techniques when using Gaussian thermal-state ansatzes. Overall, the paper provides a rigorous toolkit for understanding and exploiting the geometry of Gaussian quantum states in estimation and learning tasks.

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, Kubo-Mori, and $α$-$z$ 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.

Information geometry of bosonic Gaussian thermal states

TL;DR

This work develops the information-geometric characterization of bosonic Gaussian thermal states parameterized by their mean vector and Hamiltonian . It derives explicit formulas for the Fisher–Bures, Kubo–Mori, and - information matrices for these states, along with derivative expressions and symmetric logarithmic derivatives, enabling both multiparameter estimation bounds and single-parameter optimization strategies. The results are illustrated by a squeezed-thermal single-mode example and have potential applications in quantum metrology, gradient-based quantum machine learning, and natural gradient techniques when using Gaussian thermal-state ansatzes. Overall, the paper provides a rigorous toolkit for understanding and exploiting the geometry of Gaussian quantum states in estimation and learning tasks.

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, Kubo-Mori, and - 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.

Paper Structure

This paper contains 26 sections, 9 theorems, 112 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Motivation
  3. Main results
  4. Comparison with prior work
  5. Paper organization
  6. Preliminaries and notation
  7. Bosonic Gaussian thermal states
  8. Fisher--Bures, Kubo--Mori, and $\alpha$-$z$ information matrices
  9. General thermal states
  10. Derivative of a thermal state
  11. Fisher--Bures, Kubo--Mori, and $\alpha$-$z$ information matrices
  12. Symmetric logarithmic derivative of a thermal state
  13. Fisher--Bures, Kubo--Mori, and $\alpha$-$z$ information matrix elements of bosonic Gaussian thermal states
  14. Derivative of a bosonic Gaussian thermal state
  15. Symmetric logarithmic derivative of a bosonic Gaussian thermal state
  16. ...and 11 more sections

Key Result

Proposition 1

For $\rho(\theta)$ defined as in eq:thermal-state, the following equality holds: where the channel $\Phi_{\theta}$ is defined in eq:Phi-channel.

Figures (1)

  • Figure 1: Select Fisher--Bures information matrix elements as a function of the squeezing parameter $r\in [-1,1]$; here $\epsilon = 0.1$, $\theta = 0$ (top) and $\theta = \pi/4$ (bottom).

Theorems & Definitions (10)

  • Proposition 1: Derivative of a thermal state
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Remark 1
  • Proposition 5
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 1