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Doubly minimized Petz and sandwiched Renyi mutual information: Operational interpretation from binary quantum state discrimination

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Pith's one-line read Binary quantum discrimination for correlations shows that the direct exponent equals the doubly minimized Petz Renyi mutual information for alpha in (1/2,1) while the strong converse equals the sandwiched version for alpha greater than 1.

desk verdict The paper gives operational meanings to the doubly minimized Petz and sandwiched Renyi mutual informations as direct and strong-converse exponents in quantum correlation-detection tasks, extending the classical case. read the letter →

arxiv 2406.03213 v3 submitted 2024-06-05 quant-ph cs.ITmath.IT

Laura Burri This is my paper · ORCID
classification quant-phcs.ITmath.IT
keywords RenyimutualinformationPetzdivergencesandwichedquantumstatediscriminationerrorexponentscorrelationdetectionoperationalinterpretation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper links two minimized versions of Renyi mutual information to error rates in quantum tasks that detect whether two systems are correlated. It examines binary state discrimination problems where one must distinguish a given joint state from the closest product state. The direct exponent, which governs how fast the error probability goes to zero, matches the doubly minimized Petz version for alpha between one-half and one. The strong converse exponent, which governs the rate at which the error stays away from zero, matches the doubly minimized sandwiched version for alpha above one. The same framework also yields the Stein exponent and second-order asymptotics, extending the classical probability case to quantum states.

What carries the argument

Doubly minimized Petz and sandwiched Renyi mutual information, obtained by taking the minimum of the respective divergence over all product states, which directly set the exponential rates in correlation-detection discrimination tasks.

What would settle it

A concrete pair of quantum states and measurement strategy where the observed direct exponent in a correlation detection task differs from the value of the doubly minimized Petz Renyi mutual information at the corresponding alpha.

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Extended reading notes

Core claim

In binary quantum state discrimination problems related to correlation detection, the direct exponent is determined by the doubly minimized Petz Renyi mutual information of order α ∈ (1/2,1), and the strong converse exponent is determined by the doubly minimized sandwiched Renyi mutual information of order α ∈ (1,∞). This provides an operational interpretation of these types of Renyi mutual information and generalizes previous results for classical probability distributions to the quantum setting. For completeness, the corresponding moderate deviation regime both below and above the threshold is also studied, along with the Stein exponent and the second-order asymptotics.

Load-bearing premise

That the binary discrimination problems examined, with the minimum taken over product states, correctly isolate the correlation quantities that control the error exponents.

Editorial extensions

If this is right

  • The direct exponent in correlation detection discrimination is given exactly by the doubly minimized Petz Renyi mutual information for α in (1/2,1).
  • The strong converse exponent is given exactly by the doubly minimized sandwiched Renyi mutual information for α greater than 1.
  • Moderate deviation probabilities are characterized in both the sub- and super-threshold regimes.
  • The Stein exponent and second-order asymptotic terms are determined for the same family of tasks.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same minimization technique could be applied to other quantum divergences to obtain operational meanings in related discrimination settings.
  • These quantities may provide bounds on the resources needed to detect correlations in larger multipartite systems.
  • Classical methods for computing Renyi information exponents may transfer to the quantum case once the correct divergence is chosen.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 3 minor

Summary. The manuscript defines the doubly minimized Petz Renyi mutual information of order α as the minimum of the Petz divergence D_α(ρ || σ) over all product states σ, and defines the doubly minimized sandwiched Renyi mutual information analogously using the sandwiched divergence. It considers binary quantum state discrimination tasks for correlation detection (one joint state versus product states) and proves that the direct error exponent equals the doubly minimized Petz quantity for α ∈ (1/2,1) while the strong-converse exponent equals the doubly minimized sandwiched quantity for α ∈ (1,∞). Additional results establish the moderate-deviations rate, the Stein exponent, and second-order asymptotics, generalizing prior classical results to the quantum setting.

Significance. If the claimed exponent equalities hold, the work supplies operational meanings for these doubly minimized Renyi mutual informations via concrete discrimination tasks, extending classical correlation measures to quantum states. The inclusion of direct, strong-converse, moderate-deviation, Stein, and second-order regimes provides a comprehensive asymptotic picture. The manuscript ships explicit mathematical proofs of the exponent identifications, which is a clear strength.

minor comments (3)
  1. [Abstract] Abstract: the ranges α ∈ (1/2,1) and α ∈ (1,∞) are stated without cross-reference to the theorem that establishes the corresponding exponent equalities; adding a pointer to the relevant result would improve readability.
  2. [Section 2 or 3 (definitions)] The definition of the binary discrimination problem (hypothesis H0: joint state ρ versus H1: product states) should be stated with an explicit equation number in the main text so that later exponent claims can cite it directly.
  3. [Throughout] Notation for the doubly minimized quantities (e.g., I_α^min or similar) should be introduced once and used consistently; occasional re-use of the full descriptive phrase creates minor ambiguity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the clear summary of its contributions, and the recommendation for minor revision. No major comments were raised in the report.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: definitions precede independent operational proofs

full rationale

The doubly minimized Petz and sandwiched Renyi mutual informations are defined first as explicit minimizations of the respective divergences over all product states. The paper then proves (via direct achievability and converse arguments) that these quantities equal the direct exponent and strong-converse exponent, respectively, in independently specified binary hypothesis-testing tasks for correlation detection. No step reduces the claimed equality to a definition, a fitted parameter, or a self-citation chain; the discrimination setup and error exponents are external to the information quantities. This is the standard, non-circular route to operational interpretation.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

Based on abstract only; the work relies on standard properties of quantum Renyi divergences and asymptotic exponents in hypothesis testing.

assumptions (1)
  • standard math Petz and sandwiched Renyi divergences satisfy the necessary monotonicity and data-processing properties in the quantum setting
    Invoked implicitly when defining the minimized mutual informations and linking them to discrimination exponents.

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Cite this review

Pith. "Pith review of Doubly minimized Petz and sandwiched Renyi mutual information: Operational interpretation from binary quantum state discrimination." pith.science (2026). https://pith.science/paper/2406.03213

@misc{pith2026240603213,
  author       = {Pith},
  title        = {Pith review of: Doubly minimized Petz and sandwiched Renyi mutual information: Operational interpretation from binary quantum state discrimination},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2406.03213}},
  note         = {Machine review of arXiv:2406.03213}
}
abstract

The doubly minimized Petz Renyi mutual information of order $\alpha$ is defined as the minimum of the Petz divergence of order $\alpha$ of a given bipartite quantum state relative to all product states. The doubly minimized sandwiched Renyi mutual information is defined analogously, with the Petz divergence replaced by the sandwiched divergence. In this work, we study certain binary quantum state discrimination problems related to correlation detection. We show that the corresponding direct exponent is determined by the doubly minimized Petz Renyi mutual information of order $\alpha\in (1/2,1)$, and that the strong converse exponent is determined by the doubly minimized sandwiched Renyi mutual information of order $\alpha\in (1,\infty)$. This provides an operational interpretation of these types of Renyi mutual information and generalizes previous results for classical probability distributions to the quantum setting. For completeness, we also study the corresponding moderate deviation regime both below and above the threshold, and determine the Stein exponent and the second-order asymptotics.

Figures

Figures reproduced from arXiv: 2406.03213 by the authors.

Figure 1
Figure 1. Comparison of SRMIs for a pure state. Suppose [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗

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Forward citations

Cited by 2 Pith papers

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

  1. Alternating minimization for computing doubly minimized Petz Renyi mutual information

    quant-ph 2025-07 accept novelty 7.0 of 10

    Alternating minimization provably computes the doubly minimized Petz Rényi mutual information for all quantum states, with linear convergence for α∈(1,2] and O(1/n) convergence for α∈(1/2,1).

  2. Doubly minimized Petz and sandwiched Renyi mutual information: Properties

    quant-ph 2024-06 unverdicted novelty 7.0 of 10

    Proves additivity of doubly minimized Petz Renyi mutual information for alpha in [1/2,2] and a novel duality plus additivity for the sandwiched version for alpha in [2/3, infinity] via Sion's minimax theorem.

Reference graph

Works this paper leans on

96 extracted references · 96 canonical work pages · cited by 2 Pith papers

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    Proof of Proposition 12 We prove the listed items not in alphabetical order, but in a d ifferent order. Proof of (a), (b), (c), (e), (g), (h), (i), (j), (k), (l). These properties follow from the correspond- ing properties of the sandwiched divergence, see Propositi on 2. In particular, (e) follows from the non-negativity of the sandwiched divergence becau...

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    These properties follow from the corresponding properties of the sand- wiched divergence, see Proposition 2

    Proof of Proposition 13 Proof of (a), (e), (k), (o). These properties follow from the corresponding properties of the sand- wiched divergence, see Proposition 2. In particular, (o) follows from the additivity and positiv e definiteness of the sandwiched divergence. Proof of (h), (j), (l), (m). These properties have been proved in previous work. For (h) a n...

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    This assertion follows from the symmetry of the definition of the non-minimized SRMI in ( 2.23) with respect to A and B

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    Proof of Proposition 4 Proof of (a), (b), (c), (d), (e), (g), (h), (i), (j), (k), (l), (m), (n), (o), (p), (s). These properties follow from the corresponding properties of the minimized g eneralized SRMI since ˜I ↑↓ α (A : B)ρ = ˜I ↓ α(ρAB∥ρA), see Proposition 13. Proof of (q). By duality (d), ˜I ↑↓ α (A : B)ρ = − ˜D α 2α− 1 (ρA∥ρ− 1 A ) = 2 H 1 2α− 1 (A...

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    Proof of Theorem 5 Proof of (a). This assertion follows from the symmetry of the definition of the doubly minimized SRMI in ( 2.25) with respect to A and B. Proof of (b), (f ), (i), (k), (l), (p). Since ˜I ↓↓ α (A : B)ρ = inf σA∈S (A) ˜I ↓ α(ρAB∥σA), these properties follow from the corresponding properties of the minimized g eneralized SRMI, see Propositi...

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    Proof of Theorem 6 First, we will derive the bounds on the right-hand side of ( 3.17). Let ρAB ∈ S (AB). Then, for any R ∈ [0, ∞ ) 0 = lim s→ 1+ s − 1 s (R − ˜I ↓↓ s (A : B)ρ) ≤ sup s∈ (1,∞ ) s − 1 s (R − ˜I ↓↓ s (A : B)ρ) (E.1) 35 ≤ max(0, sup s∈ (1,∞ ) (R − ˜I ↓↓ s (A : B)ρ)) = max(0 , R − I(A : B)ρ) (E.2) due to the monotonicity and continuity of the d...

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    Let ρAB ∈ S (AB) be separable and such that ρAB ̸= ρA ⊗ ρB and I(A : B)ρ ̸= ˜I ↓↓ ∞ (A : B)ρ

    Example for Remark 3 Suppose dA ≥ 2, dB ≥ 2. Let ρAB ∈ S (AB) be separable and such that ρAB ̸= ρA ⊗ ρB and I(A : B)ρ ̸= ˜I ↓↓ ∞ (A : B)ρ. (For instance, one may consider a copy-CC state ρAB as in [ 14, Figure 2].) Consider now the left-hand side of ( 3.17) with ˆαn,ρ replaced...

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