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arxiv: 2601.13970 · v2 · pith:XCKOHVB4new · submitted 2026-01-20 · 🪐 quant-ph

A Converse Bound via the Nussbaum-Szko{l}a Mapping for Quantum Hypothesis Testing

Jorge Lizarribar-Carrillo , Gonzalo Vazquez-Vilar , Tobias Koch This is my paper

Pith reviewed 2026-05-21 15:23 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum hypothesis testingconverse boundNussbaum-Szkoła mappingasymptotic regimesfinite blocklengtherror trade-offquantum state discrimination
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The pith

A lower bound from the Nussbaum-Szkoła mapping unifies converse results for asymmetric quantum hypothesis testing across all regimes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper develops a new lower bound on the trade-off between the two types of error probabilities when trying to distinguish which of two quantum states is present. The bound is constructed by applying the Nussbaum-Szkoła mapping to turn the quantum problem into a classical hypothesis-testing problem. From one expression the bound recovers the known optimal exponents in the large-deviation, moderate-deviation, and small-deviation asymptotic regimes. It also supplies useful numerical approximations for short sequences of states and is tighter than earlier bounds that rely on fidelity or information-spectrum techniques. A reader would care because the single-formula approach removes the need for separate technical arguments for each regime and improves estimates of fundamental performance limits in practical quantum state discrimination.

Core claim

The paper shows that the Nussbaum-Szkoła mapping applied to asymmetric quantum hypothesis testing produces a converse bound that holds uniformly and recovers the optimal error exponents in the large-, moderate-, and small-deviation regimes from a single expression, while also furnishing accurate approximations to the optimal finite-blocklength error trade-off curve.

What carries the argument

The Nussbaum-Szkoła mapping, which converts a pair of quantum states into a pair of classical probability distributions while preserving the quantities relevant to hypothesis testing error exponents.

Load-bearing premise

The Nussbaum-Szkoła mapping can be applied directly to the asymmetric quantum hypothesis testing problem to yield a valid unified lower bound without extra regime-specific adjustments.

What would settle it

Compute the exact optimal error-probability trade-off curve for a concrete pair of quantum states at small blocklength n and check whether the new bound lies below the true curve yet closer to it than the fidelity or information-spectrum bounds, or verify that the bound exactly reproduces the known quantum Chernoff exponent in the large-deviation limit.

Figures

Figures reproduced from arXiv: 2601.13970 by Gonzalo Vazquez-Vilar, Jorge Lizarribar-Carrillo, Tobias Koch.

Figure 1
Figure 1. Figure 1: Quantum error trade-off compared to the lower bounds from 55 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Upper bounds on the normalized hypothesis testing relative entropy [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
read the original 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-Szko{\l}a 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces a novel lower bound for asymmetric quantum hypothesis testing derived from the Nussbaum-Szkoła mapping. The bound is presented as arising from a single expression that recovers known converse results across large-deviation, moderate-deviation, and small-deviation regimes, permits direct application of classical results in some cases, and yields tighter numerical approximations to the optimal error trade-off at small blocklengths than fidelity or information-spectrum bounds.

Significance. If the unification claim holds without regime-specific looseness, the result would provide a practical, unified tool for converse bounds in quantum hypothesis testing, bridging asymptotic regimes and improving finite-blocklength estimates. The potential to leverage classical results directly would be a notable strength.

major comments (2)
  1. [§3] §3 (main theorem): The specialization of the single bound expression to the known tight converses (e.g., Chernoff-Stein lemma in large deviations, moderate-deviation rate functions) must be shown explicitly via parameter limits or substitutions, without auxiliary information-spectrum arguments or regime-dependent adjustments, to substantiate the 'unified recovery from one expression' claim.
  2. [§4] §4 (asymptotic recovery): The algebra demonstrating recovery of all three regimes should be presented in full, including any intermediate steps that might introduce looseness when the mapping is applied to quantum states rather than classical distributions.
minor comments (2)
  1. [Numerical results] Figure 1 and Table 1: Ensure axis labels and captions explicitly state the quantum states (e.g., qubit Werner states) and blocklength values used in the numerical comparisons for reproducibility.
  2. [Preliminaries] Notation section: Define the precise form of the Nussbaum-Szkoła mapping when lifted to quantum hypothesis testing, including how the classical probability measures are obtained from the quantum states.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and constructive feedback on our manuscript. The comments highlight important aspects of our unification claim, and we address them point by point below. We have revised the manuscript to incorporate explicit derivations as requested.

read point-by-point responses

  1. Referee: [§3] §3 (main theorem): The specialization of the single bound expression to the known tight converses (e.g., Chernoff-Stein lemma in large deviations, moderate-deviation rate functions) must be shown explicitly via parameter limits or substitutions, without auxiliary information-spectrum arguments or regime-dependent adjustments, to substantiate the 'unified recovery from one expression' claim.

    Authors: We agree that explicit specialization is essential to substantiate the claim of recovery from a single expression. In the revised version, we add a dedicated subsection to §3 that derives the Chernoff-Stein lemma and the moderate-deviation rate function directly from the bound via parameter limits and substitutions. These derivations use only the definition of the Nussbaum-Szkoła mapping and standard limit arguments, without invoking information-spectrum methods or regime-specific adjustments. revision: yes

  2. Referee: [§4] §4 (asymptotic recovery): The algebra demonstrating recovery of all three regimes should be presented in full, including any intermediate steps that might introduce looseness when the mapping is applied to quantum states rather than classical distributions.

    Authors: We accept this recommendation. The revised §4 now contains the complete algebraic steps for recovering the large-, moderate-, and small-deviation regimes. We explicitly track each intermediate inequality arising from the quantum-to-classical mapping and demonstrate that any looseness is confined to the inherent properties of the bound itself rather than additional artifacts from the mapping. This expansion clarifies the tightness in each asymptotic regime. revision: yes

Circularity Check

0 steps flagged

No significant circularity; bound derived from external Nussbaum-Szkoła mapping

full rationale

The derivation applies the established Nussbaum-Szkoła mapping to asymmetric quantum hypothesis testing to obtain a single lower-bound expression. This mapping originates from prior independent classical work and is not redefined or fitted within the paper. The abstract states that the bound recovers converses across regimes directly from this expression without auxiliary information-spectrum arguments or regime-specific adjustments. No equations reduce by construction to fitted inputs, self-citations, or ansatzes imported from the authors' own prior results. The central claim remains independent of the target result and is supported by the mapping's properties plus classical results, making the paper self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; the bound is described as arising from an existing mapping applied to quantum hypothesis testing.

pith-pipeline@v0.9.0 · 5676 in / 1082 out tokens · 122395 ms · 2026-05-21T15:23:10.642296+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Multiple Quantum Hypothesis Testing: One-Shot Pairwise Bounds and Sharp Asymptotics

    quant-ph 2026-06 unverdicted novelty 7.0

    Establishes dimension-free one-shot pairwise bounds for multiple quantum hypothesis testing, resolves Audenaert-Mosonyi conjecture, and proves achievability of multiple quantum Chernoff distance for arbitrary separabl...

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages · cited by 1 Pith paper

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