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Error exponents of quantum state discrimination with composite correlated hypotheses

Kun Fang, Masahito Hayashi

TL;DR

The paper advances quantum hypothesis testing by extending error-exponent and strong-converse analyses from i.i.d. to composite, correlated sets of quantum states. It introduces and interrelates two natural extensions of Hoeffding-type divergences to sets, and proves that the error exponent is governed by the regularized Hoeffding divergence while the strong-converse exponent is bounded below by the regularized Hoeffding anti-divergence with a matching upper bound in the singleton-null case. These results yield a sharp asymptotic trade-off between Type-I and Type-II errors for composite hypotheses and enable applications to adversarial channel discrimination and quantum resource detection. The framework recovers and strengthens generalized Stein lemmas for sets, and it exposes discontinuities of regularized Petz Rényi divergences, illuminating limitations of continuity in set-based settings. Collectively, the work provides a versatile toolkit for error exponents in robust quantum discrimination tasks with broad implications for quantum information theory and resource theories.

Abstract

We study the error exponents in quantum hypothesis testing between two sets of quantum states, extending the analysis beyond the independent and identically distributed case to encompass composite correlated hypotheses. In particular, we introduce and compare two natural extensions of the quantum Hoeffding divergence and anti-divergence to sets of quantum states, establishing their equivalence or quantitative relations. In the error exponent regime, we generalize the quantum Hoeffding bound to stable sequences of convex, compact sets of quantum states, demonstrating that the optimal Type-I error exponent, under an exponential constraint on the Type-II error, is precisely characterized by the regularized quantum Hoeffding divergence between the sets. In the strong converse exponent regime, we provide a general lower bound on the exponent in terms of the regularized quantum Hoeffding anti-divergence and a matching upper bound when the null hypothesis is a singleton. The generality of these results enables applications in various contexts, including (i) refining the generalized quantum Stein's lemma by [Fang, Fawzi & Fawzi, 2024]; (ii) exhibiting counterexamples to the continuity of the regularized Petz Rényi divergence and Hoeffding divergence; (iii) obtaining error exponents for adversarial channel discrimination and resource detection problems.

Error exponents of quantum state discrimination with composite correlated hypotheses

TL;DR

The paper advances quantum hypothesis testing by extending error-exponent and strong-converse analyses from i.i.d. to composite, correlated sets of quantum states. It introduces and interrelates two natural extensions of Hoeffding-type divergences to sets, and proves that the error exponent is governed by the regularized Hoeffding divergence while the strong-converse exponent is bounded below by the regularized Hoeffding anti-divergence with a matching upper bound in the singleton-null case. These results yield a sharp asymptotic trade-off between Type-I and Type-II errors for composite hypotheses and enable applications to adversarial channel discrimination and quantum resource detection. The framework recovers and strengthens generalized Stein lemmas for sets, and it exposes discontinuities of regularized Petz Rényi divergences, illuminating limitations of continuity in set-based settings. Collectively, the work provides a versatile toolkit for error exponents in robust quantum discrimination tasks with broad implications for quantum information theory and resource theories.

Abstract

We study the error exponents in quantum hypothesis testing between two sets of quantum states, extending the analysis beyond the independent and identically distributed case to encompass composite correlated hypotheses. In particular, we introduce and compare two natural extensions of the quantum Hoeffding divergence and anti-divergence to sets of quantum states, establishing their equivalence or quantitative relations. In the error exponent regime, we generalize the quantum Hoeffding bound to stable sequences of convex, compact sets of quantum states, demonstrating that the optimal Type-I error exponent, under an exponential constraint on the Type-II error, is precisely characterized by the regularized quantum Hoeffding divergence between the sets. In the strong converse exponent regime, we provide a general lower bound on the exponent in terms of the regularized quantum Hoeffding anti-divergence and a matching upper bound when the null hypothesis is a singleton. The generality of these results enables applications in various contexts, including (i) refining the generalized quantum Stein's lemma by [Fang, Fawzi & Fawzi, 2024]; (ii) exhibiting counterexamples to the continuity of the regularized Petz Rényi divergence and Hoeffding divergence; (iii) obtaining error exponents for adversarial channel discrimination and resource detection problems.

Paper Structure

This paper contains 39 sections, 21 theorems, 264 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Quantum hypothesis testing between two quantum states
  3. Stein exponent.
  4. Error expoent.
  5. Strong converse exponent.
  6. Quantum hypothesis testing between two sets of quantum states
  7. Summary of main results
  8. Error exponent.
  9. Strong converse exponent.
  10. Applications.
  11. Organization of the paper
  12. Preliminaries
  13. Notations
  14. Quantum divergences
  15. Hoeffding divergence and anti-divergence for sets of states
  16. ...and 24 more sections

Key Result

Lemma 3.1

For any $\rho_m, \sigma_m \in \mathscr{D}({\cal H}^{\otimes m})$ and $\rho_n, \sigma_n \in \mathscr{D}({\cal H}^{\otimes n})$, it holds that

Figures (10)

  • Figure 1: Illustration of different study regimes of quantum hypothesis testing. Each curve represents the tradeoff between the Type-I and Type-II errors for varying block lengths $n$, with darker lines corresponding to longer block lengths. (I) represents the Stein exponent regime, (II) represents the error exponent regime, and (III) represents the strong converse exponent regime.
  • Figure 2: Quantum hypothesis testing between two sets of quantum states ${{\mathscr{A}}}_n$ and ${{\mathscr{B}}}_n$.
  • Figure 3: Summary of results in the error exponent regime. Quantitative relationships between the various quantities are indicated by black arrows: the quantity at the tail of an arrow is always greater than or equal to the one at the head. A double arrow indicates an equality.
  • Figure 4: Summary of results in the strong converse exponent regime. Quantitative relationships between the various quantities are indicated by black arrows: the quantity at the tail of an arrow is always greater than or equal to the one at the head. A double arrow indicates an equality. A dashed arrow indicates partial progress when ${{\mathscr{A}}}_n$ is a singleton.
  • Figure : (a) Petz Rényi divergence
  • ...and 5 more figures

Theorems & Definitions (64)

  • Definition 2.1: Umegaki relative entropy umegaki1954conditional.
  • Definition 2.2: Petz Rényi divergence petz1986quasi.
  • Definition 2.3: Sandwiched Rényi divergence muller2013quantumwilde2014strong.
  • Definition 2.4: Quantum divergence between two sets of states.
  • Remark 2.1
  • Definition 2.5: Stable sequence.
  • Remark 2.2
  • Definition 3.1
  • Lemma 3.1: Subadditivity.
  • proof
  • ...and 54 more