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Sample Complexity of Locally Differentially Private Quantum Hypothesis Testing

Hao-Chung Cheng, Christoph Hirche, Cambyse Rouzé

TL;DR

This work gives achievability and converse bounds that nearly match the best known classical bounds on quantum state discrimination and proves several novel inequalities between quantum divergences that should be of independent interest.

Abstract

Quantum state discrimination is an important problem in many information processing tasks. In this work we are concerned with finding its best possible sample complexity when the states are preprocessed by a quantum channel that is required to be locally differentially private. To that end we provide achievability and converse bounds for different settings. This includes symmetric state discrimination in various regimes and the asymmetric case. On the way, we also prove new sample complexity bounds for the general unconstrained setting. An important tool in this endeavor are new entropy inequalities that we believe to be of independent interest.

Sample Complexity of Locally Differentially Private Quantum Hypothesis Testing

TL;DR

This work gives achievability and converse bounds that nearly match the best known classical bounds on quantum state discrimination and proves several novel inequalities between quantum divergences that should be of independent interest.

Abstract

Quantum state discrimination is an important problem in many information processing tasks. In this work we are concerned with finding its best possible sample complexity when the states are preprocessed by a quantum channel that is required to be locally differentially private. To that end we provide achievability and converse bounds for different settings. This includes symmetric state discrimination in various regimes and the asymmetric case. On the way, we also prove new sample complexity bounds for the general unconstrained setting. An important tool in this endeavor are new entropy inequalities that we believe to be of independent interest.
Paper Structure (19 sections, 24 theorems, 141 equations)

This paper contains 19 sections, 24 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Divergences and distance measures
  5. Sample complexity of hypothesis testing
  6. Proof of Theorem \ref{['Thm:SC-ap']}
  7. Locally differentially private hypothesis testing
  8. Symmetric hypothesis testing with uniform prior
  9. Achieving LDP optimal sample complexity
  10. Optimality of LDP sample complexity
  11. Converse, Part I
  12. Converse, Part II
  13. Symmetric hypothesis testing with arbitrary prior
  14. Asymmetric hypothesis testing
  15. Proofs
  16. ...and 4 more sections

Key Result

Lemma 3.1

For any two states $\rho,\sigma\in\mathcal{S}_d$,

Theorems & Definitions (47)

  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Theorem 3.3
  • proof
  • Lemma 3.4
  • proof
  • Proposition 3.5
  • proof
  • ...and 37 more