Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Generating functions for quantum metric, Berry curvature, and quantum Fisher information matrix

Wei Chen

TL;DR

This work establishes fidelity between density matrices as a generating function for quantum information geometry quantities, enabling the extraction of the quantum Fisher information matrix $F_{μν}$ and the Christoffel symbols $Γ_{λμν}$ via parameter derivatives. In the pure-state limit, the phase of the state product generates the Berry curvature while the modulus generates the quantum metric, linking to a broader information-geometry framework in the real-wave-function limit where the classical Fisher information emerges. The authors provide explicit Bloch-2×2 expressions for the generating functions, and demonstrate the formalism with canonical two-level systems at finite temperature and the Su–Schrieffer–Heeger model, showing how finite temperature mitigates quantum geometry. This work thus unifies mixed-state quantum geometry, pure-state quantum geometry, and information geometry into a single generating-function perspective with practical computational tools for two-level systems. The results have potential implications for quantum metrology and finite-temperature topological materials by encoding all geometric information in the generating function.

Abstract

We elaborate that the fidelity between two density matrices is a generating function, through which the quantum Fisher information matrix and Christoffel symbol of the first kind in the parameter space can be obtained through derivatives with respect to the parameters. For pure states, the fidelity and phase of the product between two quantum states are shown to be the generating functions of the quantum metric and Berry curvature, respectively. Further limiting to systems described by real wave functions, our formalism recovers the well-known result that the fidelity between two probability mass functions is the generating function of the classical Fisher information matrix, indicating a hierarchy of quantum to information geometry. The Bloch representation of the generating functions is given explicitly for $2\times 2$ density matrices, and the application to canonical ensemble of Su-Schrieffer-Heeger model suggests the mitigation of quantum geometry at finite temperature.

Generating functions for quantum metric, Berry curvature, and quantum Fisher information matrix

TL;DR

This work establishes fidelity between density matrices as a generating function for quantum information geometry quantities, enabling the extraction of the quantum Fisher information matrix and the Christoffel symbols via parameter derivatives. In the pure-state limit, the phase of the state product generates the Berry curvature while the modulus generates the quantum metric, linking to a broader information-geometry framework in the real-wave-function limit where the classical Fisher information emerges. The authors provide explicit Bloch-2×2 expressions for the generating functions, and demonstrate the formalism with canonical two-level systems at finite temperature and the Su–Schrieffer–Heeger model, showing how finite temperature mitigates quantum geometry. This work thus unifies mixed-state quantum geometry, pure-state quantum geometry, and information geometry into a single generating-function perspective with practical computational tools for two-level systems. The results have potential implications for quantum metrology and finite-temperature topological materials by encoding all geometric information in the generating function.

Abstract

We elaborate that the fidelity between two density matrices is a generating function, through which the quantum Fisher information matrix and Christoffel symbol of the first kind in the parameter space can be obtained through derivatives with respect to the parameters. For pure states, the fidelity and phase of the product between two quantum states are shown to be the generating functions of the quantum metric and Berry curvature, respectively. Further limiting to systems described by real wave functions, our formalism recovers the well-known result that the fidelity between two probability mass functions is the generating function of the classical Fisher information matrix, indicating a hierarchy of quantum to information geometry. The Bloch representation of the generating functions is given explicitly for density matrices, and the application to canonical ensemble of Su-Schrieffer-Heeger model suggests the mitigation of quantum geometry at finite temperature.

Paper Structure

This paper contains 11 sections, 49 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Mixed state formalism
  3. Fidelity as a generating function for QFIM
  4. Bloch representation of $2\times 2$ density matrices
  5. Pure state limit
  6. Real wave function limit
  7. Applications
  8. Canonical ensemble of two-level systems
  9. Su-Schrieffer-Heeger model at finite temperature
  10. Conclusions
  11. Acknowledgments

Figures (3)

  • Figure 1: Summary of the hierarchy of quantum geometry, where the most general mixed state quantum geometry descends to the pure state quantum geometry for pure state systems, which further descends to the information geometry if the wave function of the pure state is real. The metric and the fidelity as a generating function both have their correspondence in all three levels of the hierarchy. The present work contributes to the pure and mixed state generating functions. The logarithm inside the parenthesis means that the fidelity with or without the logarithm can serve as a generating function.
  • Figure 2: (a) The fidelity ${\rm Tr}\sqrt{\sqrt{\rho(B)}\rho(B')\sqrt{\rho(B)}}$ as the generating function for the QFIM for a single spin $1/2$ at finite temperature, plotted as a function of the dimentionless parameters $\mu_{B}B/k_{B}T$ and $\mu_{B}B'/k_{B}T$. (b) The QFIM $F_{BB}$ in units of $(\mu_{B}/k_{B}T)^{2}$ and the negative of the Christoffel symbol $-\Gamma_{BBB}$ in units of $(\mu_{B}/k_{B}T)^{3}$ plotted as a function of $\mu_{B}B/k_{B}T$.
  • Figure 3: The generating function $|\tilde{B}(k,k')|={\rm Tr}\sqrt{\sqrt{\rho(k)}\rho(k')\sqrt{\rho(k)}}$ of QFIM, or equivalently the fidelity between density matrices, for the SSH model at (a) $T=0$ and (b) $T=0.5$ plotted as a function of the two momenta $k$ and $k'$, when the hopping difference takes the value $\delta t=0.2$. Red color corresponds to the maximum value of 1, and violet corresponds to the minimal value of 0.77. (c) The QFIM $F_{kk}$ and (d) the Christoffel symbol $\Gamma_{kkk}$ as functions of $k$ plotted for several temperatures ranging from $T=0$ to $T=0.5$.