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A Converse Bound via the Nussbaum-Szkoła Mapping for Quantum Hypothesis Testing

Jorge Lizarribar-Carrillo, Gonzalo Vazquez-Vilar, Tobias Koch

Abstract

Quantum hypothesis testing concerns the discrimination between quantum states. This paper introduces a novel lower bound for asymmetric quantum hypothesis testing that is based on the Nussbaum-Szkoła mapping. The lower bound provides a unified recovery of converse results across all major asymptotic regimes, including large-, moderate-, and small-deviations. Unlike existing bounds, which either rely on technically involved information-spectrum arguments or suffer from fixed prefactors and limited applicability in the non-asymptotic regime, the proposed bound arises from a single expression and enables, in some cases, the direct use of classical results. It is further demonstrated that the proposed bound provides accurate approximations to the optimal quantum error trade-off function at small blocklengths. Numerical comparisons with existing bounds, including those based on fidelity and information spectrum methods, highlight its improved tightness and practical relevance.

A Converse Bound via the Nussbaum-Szkoła Mapping for Quantum Hypothesis Testing

Abstract

Quantum hypothesis testing concerns the discrimination between quantum states. This paper introduces a novel lower bound for asymmetric quantum hypothesis testing that is based on the Nussbaum-Szkoła mapping. The lower bound provides a unified recovery of converse results across all major asymptotic regimes, including large-, moderate-, and small-deviations. Unlike existing bounds, which either rely on technically involved information-spectrum arguments or suffer from fixed prefactors and limited applicability in the non-asymptotic regime, the proposed bound arises from a single expression and enables, in some cases, the direct use of classical results. It is further demonstrated that the proposed bound provides accurate approximations to the optimal quantum error trade-off function at small blocklengths. Numerical comparisons with existing bounds, including those based on fidelity and information spectrum methods, highlight its improved tightness and practical relevance.
Paper Structure (9 sections, 7 theorems, 48 equations, 2 figures)

This paper contains 9 sections, 7 theorems, 48 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Error Probability Lower Bound
  4. The Lower Bound
  5. Pure-State Discrimination
  6. Classical test
  7. Quantum test
  8. Asymptotic Analysis
  9. Comparison with Previous Converse Bounds

Key Result

Lemma 1

Let $\rho$ and $\sigma$ be two density operators. Then, where $\Pi_t \triangleq \{ t\rho - \sigma \succeq 0\}$ is the projection onto the non-negative eigenspace of $t\rho - \sigma$ and $\bar{\Pi}_t = I - \Pi_t$.

Figures (2)

  • Figure 1: Quantum error trade-off compared to the lower bounds from Theorem \ref{['thm:low']} for a hypothesis test between the states ${\ket{0}\!\bra{0}}^{\otimes 5}$ and ${\ket{+}\!\bra{+}}^{\otimes 5}$.
  • Figure 2: Upper bounds on the normalized hypothesis testing relative entropy $\frac{1}{n}D_h^{\varepsilon}(\rho^{\otimes n} \| \sigma^{\otimes n})$ for $\rho,\sigma$ in \ref{['eq:mixedstates']} and $n=5$.

Theorems & Definitions (7)

  • Lemma 1: vazquez2016multiple
  • Theorem 1
  • Lemma 2
  • Corollary 1: Small Deviations
  • Corollary 2: Moderate Deviations
  • Lemma 3: rozovsky2002estimate
  • Corollary 3: Large Deviations