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Generalized quantum Chernoff bound

Kun Fang, Masahito Hayashi

TL;DR

This work extends the quantum Chernoff bound from simple hypothesis testing to discriminating among multiple sets of quantum states, including composite and correlated scenarios. Under convexity and tensor-product stability, the authors show that the optimal exponential error rate equals the regularized Chernoff divergence between the sets, and establish a minimax principle guaranteeing a state-agnostic optimal test. They provide explicit constructions for binary composite tests and a detailed treatment of two-set and multi-set Chernoff bounds, including a constructive spectrum-discretization approach for the lower bound. An operational interpretation is given for the maximum overlap with free states in quantum resource theories, linking hypothesis testing performance to resource quantify. The results unify and generalize classical and quantum Chernoff bounds and have implications for adversarial quantum hypothesis testing and resource-theoretic distinguishability.

Abstract

We consider the task of distinguishing whether a quantum system is prepared in a state from one of several sets of quantum states. Assuming their convexity and stability under tensor product, we prove that the optimal error exponent for discrimination is precisely given by the regularized quantum Chernoff divergence between the sets, thereby establishing a generalized quantum Chernoff bound for the discrimination of multiple sets of quantum states. This extends the classical and quantum Chernoff bounds to the general setting of composite and correlated quantum hypotheses. Furthermore, leveraging minimax theorems, we show that discriminating between sets of quantum states is no harder than discriminating between their worst-case elements in terms of error probability. This implies the existence of an optimal state-agnostic test that achieves the minimum error probability for all states in the sets, matching the performance of the optimal state-dependent test for the most difficult pair of states. We provide explicit characterizations of the optimal state-agnostic test in the binary composite case. Finally, we show that the maximum overlap between a pure state and a set of free states, a quantity that frequently arises in quantum resource theories, is equal to the quantum Chernoff divergence between the sets, thereby providing an operational interpretation of this quantity in the context of symmetric hypothesis testing.

Generalized quantum Chernoff bound

TL;DR

This work extends the quantum Chernoff bound from simple hypothesis testing to discriminating among multiple sets of quantum states, including composite and correlated scenarios. Under convexity and tensor-product stability, the authors show that the optimal exponential error rate equals the regularized Chernoff divergence between the sets, and establish a minimax principle guaranteeing a state-agnostic optimal test. They provide explicit constructions for binary composite tests and a detailed treatment of two-set and multi-set Chernoff bounds, including a constructive spectrum-discretization approach for the lower bound. An operational interpretation is given for the maximum overlap with free states in quantum resource theories, linking hypothesis testing performance to resource quantify. The results unify and generalize classical and quantum Chernoff bounds and have implications for adversarial quantum hypothesis testing and resource-theoretic distinguishability.

Abstract

We consider the task of distinguishing whether a quantum system is prepared in a state from one of several sets of quantum states. Assuming their convexity and stability under tensor product, we prove that the optimal error exponent for discrimination is precisely given by the regularized quantum Chernoff divergence between the sets, thereby establishing a generalized quantum Chernoff bound for the discrimination of multiple sets of quantum states. This extends the classical and quantum Chernoff bounds to the general setting of composite and correlated quantum hypotheses. Furthermore, leveraging minimax theorems, we show that discriminating between sets of quantum states is no harder than discriminating between their worst-case elements in terms of error probability. This implies the existence of an optimal state-agnostic test that achieves the minimum error probability for all states in the sets, matching the performance of the optimal state-dependent test for the most difficult pair of states. We provide explicit characterizations of the optimal state-agnostic test in the binary composite case. Finally, we show that the maximum overlap between a pure state and a set of free states, a quantity that frequently arises in quantum resource theories, is equal to the quantum Chernoff divergence between the sets, thereby providing an operational interpretation of this quantity in the context of symmetric hypothesis testing.

Paper Structure

This paper contains 33 sections, 17 theorems, 150 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Quantum Chernoff bound between two quantum states.
  3. Quantum Chernoff bound for multiple quantum states.
  4. Quantum Chernoff bound for multiple sets of quantum states.
  5. Summary of main results.
  6. Organization of the paper.
  7. Preliminaries
  8. Notations
  9. Minimax theorems
  10. Useful lemmas
  11. Quantum Chernoff divergence between sets of quantum states
  12. Optimal error probability and optimal test for sets of states
  13. Optimal average error probability
  14. Optimal test for binary composite hypotheses
  15. Minimax optimization and saddle points
  16. ...and 18 more sections

Key Result

Lemma 2.1

A pair of solutions $(x^*, y^*)$ is a saddle point of $f$ if and only if the minimax equality holds and $x^*$ is an optimal solution of the problem while $y^*$ is an optimal solution of the problem

Figures (4)

  • Figure 1: Quantum hypothesis testing for multiple sets of quantum states ${{\mathscr{C}}}_{1,n}, \cdots, {{\mathscr{C}}}_{r,n}$.
  • Figure : (a) Petz Rényi divergence
  • Figure : (a) Petz Rényi divergence
  • Figure : (b) Chernoff divergence

Theorems & Definitions (39)

  • Definition 2.1
  • Remark 2.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • ...and 29 more