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Limit Distribution Theory for Quantum Divergences

Sreejith Sreekumar, Mario Berta

Abstract

Estimation of quantum relative entropy and its Rényi generalizations is a fundamental statistical task in quantum information theory, physics, and beyond. While several estimators of these divergences have been proposed in the literature along with their computational complexities explored, a limit distribution theory which characterizes the asymptotic fluctuations of the estimation error is still premature. As our main contribution, we characterize these asymptotic distributions in terms of Fréchet derivatives of elementary operator-valued functions. We achieve this by leveraging an operator version of Taylor's theorem and identifying the regularity conditions needed. As an application of our results, we consider an estimator of quantum relative entropy based on Pauli tomography of quantum states and show that the resulting asymptotic distribution is a centered normal, with its variance characterized in terms of the Pauli operators and states. We utilize the knowledge of the aforementioned limit distribution to obtain asymptotic performance guarantees for a multi-hypothesis testing problem.

Limit Distribution Theory for Quantum Divergences

Abstract

Estimation of quantum relative entropy and its Rényi generalizations is a fundamental statistical task in quantum information theory, physics, and beyond. While several estimators of these divergences have been proposed in the literature along with their computational complexities explored, a limit distribution theory which characterizes the asymptotic fluctuations of the estimation error is still premature. As our main contribution, we characterize these asymptotic distributions in terms of Fréchet derivatives of elementary operator-valued functions. We achieve this by leveraging an operator version of Taylor's theorem and identifying the regularity conditions needed. As an application of our results, we consider an estimator of quantum relative entropy based on Pauli tomography of quantum states and show that the resulting asymptotic distribution is a centered normal, with its variance characterized in terms of the Pauli operators and states. We utilize the knowledge of the aforementioned limit distribution to obtain asymptotic performance guarantees for a multi-hypothesis testing problem.
Paper Structure (27 sections, 12 theorems, 159 equations)

This paper contains 27 sections, 12 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. Related Work
  3. Paper Organization
  4. Preliminaries
  5. Notation
  6. Weak convergence of Random Density Operators
  7. Fréchet Differentiability
  8. Bochner Integrability
  9. Quantum Information Measures
  10. Main Results
  11. Quantum Relative Entropy
  12. Quantum Rényi Divergence
  13. Generalization to Infinite-dimensional Quantum Systems
  14. Application
  15. Tomographic Estimator of Quantum States
  16. ...and 12 more sections

Key Result

Theorem 1

Let $\rho_n \ll \sigma_n \ll \sigma$ and $\rho_n \ll \rho \ll \sigma$. The following hold:

Theorems & Definitions (17)

  • Definition 1: Fréchet differentiability, see e.g. Bhatia-book
  • Theorem 1: Limit distribution for quantum relative entropy
  • Remark 2: One-sample null and alternative
  • Corollary 3: Commutative case
  • Corollary 4: Limit distribution for von Neumann entropy
  • proof
  • Theorem 5: Limit distribution for Petz-Rényi divergence
  • Corollary 6: Commutative case
  • Theorem 7: Limit distribution for sandwiched Rényi divergence
  • Corollary 8: Fidelity and max-divergence
  • ...and 7 more