Information geometry of bosonic Gaussian thermal states

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 $\alpha$-$z$ 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.

arXiv commentv3: 25 pages, 1 figure, final version accepted for publication in Physical Review A; also, see related work in arXiv:2410.12935 and arXiv:2410.24058

References in corpus (37)

• Observation of Gravitational Waves from a Binary Black Hole Merger
• Quantum Machine Learning
• Gaussian Quantum Information
• Enhancing the sensitivity of the LIGO gravitational wave detector by using squeezed states of light
• Continuous variable quantum information: Gaussian states and beyond
• Quantum Fisher information matrix and multiparameter estimation
• Quantum Natural Gradient
• Quantum information with Gaussian states
• Quantum fidelity for arbitrary Gaussian states
• A Geometric Perspective on Quantum Parameter Estimation
• Teleportation of Nonclassical Wave Packets of light
• Quantum Fisher information for states in exponential form
• On quantumness in multi-parameter quantum estimation
• alpha-z-relative Renyi entropies
• Multiparameter Gaussian Quantum Metrology
• Sample-efficient learning of quantum many-body systems
• Fisher Information in Noisy Intermediate-Scale Quantum Applications
• Quantum Belief Propagation
• Estimation of Gaussian quantum states
• Gaussian quantum resource theories
• Gaussian hypothesis testing and quantum illumination
• Quantum Semiparametric Estimation
• Bounds on Quantum Multiple-Parameter Estimation with Gaussian State
• Fidelity for Multimode Thermal Squeezed States
• Information geometry of Gaussian channels
• From Wigner-Yanase-Dyson conjecture to Carlen-Frank-Lieb conjecture (New title)
• Optimal estimation with quantum optomechanical systems in the nonlinear regime
• Quantum Fisher information on two manifolds of two-mode Gaussian states
• The entropy gain of infinite-dimensional quantum channels
• Gaussian relative entropy of entanglement
• On the Sample Complexity of Quantum Boltzmann Machine Learning
• Fundamental limits of metrology at thermal equilibrium
• Estimation of Hamiltonian parameters from thermal states
• Geometric approach to quantum statistical inference
• Learning quantum states of continuous variable systems
• Limited quantum advantage for stellar interferometry via continuous-variable teleportation
• Optimal estimates of trace distance between bosonic Gaussian states and applications to learning