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REVIEW 3 major objections 3 minor 44 references

On Shor's conjecture on the accessible information of quantum dichotomies

T0 review · 3 major / 3 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read More guessable does not mean more informative, even in the binary case

desk verdict Useful disproof of monotonicity, but Proposition 2 is false as stated due to a missing same-marginal hypothesis; still worth refereeing after a fix. read the letter →

arxiv 2512.11233 v1 pith:RYFQ2R4L submitted 2025-12-12 quant-ph

Khac Duc An Thai , Michele Dall'Arno This is my paper
classification quant-ph MSC 81P4594A17 PACS 03.67.-a
keywords accessibleinformationquantumdichotomymutualguessingprobabilityextremalmeasurementsqubittestingregionmonotonicity
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

This paper tries to settle whether the accessible information of a two-state quantum encoding is always attained by an orthogonal (von Neumann) measurement, but it does so by first closing a tempting side door. The side door was the idea that if mutual information always increased with the guessing probability, then the measurement that best guesses the state would also maximize information. The authors prove this monotonicity fails except for a zero-measure set of binary distributions, and they give the exact boundary conditions. They then characterize, in closed form, the von Neumann measurements that generate extremal conditional distributions for any qubit dichotomy, reducing the search space for the accessible information to a finite interval and sharpening a standing conjecture.

What carries the argument

Two parameterizations do the work. For binary distributions, p(a,b,λ) expands the joint distribution in an orthonormal matrix basis; Lemma 2 identifies the λ-convexity, the unique minimum at λ=ab, and the fact that for fixed λ the mutual information is maximized at b=λ+a−1 on the boundary. For quantum dichotomies, the operator family H(λ)=λω−(ρ−σ), with ω the affine combination of ρ and σ orthogonal to ρ−σ, sweeps out the extreme effects; the identity Tr ω^2 < 1 yields the explicit λ* that bounds the interval of extremal measurements.

What would settle it

Verify Proposition 2 by applying the paper's construction to an interior distribution (e.g., a=0.2, b=0.3, λ=0.1): the produced pX,Z must have lower guessing probability but higher mutual information, and for a boundary case satisfying the two conditions, no binary pX,Z may violate Eq. (4). A single failure on either side refutes the characterization.

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Extended reading notes

Core claim

The paper's central claim is that 'higher guessing probability implies higher mutual information' is false for binary random variables except on a negligible set. A distribution pX,Y satisfies that implication for every binary pX,Z if and only if its trace is at least pX=0 and its guessing probability equals pY=0 + (1−pX=0); these conditions force at least one zero entry, so the set has zero measure. Second, for a qubit dichotomy (ρ,σ), the maximum of any convex objective over von Neumann measurements is attained at one of the projectors of H(λ)=λω−(ρ−σ) with λ in [−λ*,λ*], where λ* is an explicit function of the purities and overlap of ρ and σ. This tightens a previous bound and reframes a

Load-bearing premise

The paper's stronger claims stand on an imported characterization of extremal distributions — that every extreme point of a qubit dichotomy's testing region comes from a projector of H(λ) — and on an unproved monotonicity of the mutual information along a specific boundary curve; if either is wrong, the corresponding main result fails.

Editorial extensions

If this is right

  • If the Proposition 2 characterization is right, any proof of the conjecture cannot rely on a guessing-probability monotonicity; the two quantities decouple for almost all distributions.
  • For qubit dichotomies, computing accessible information can be restricted to von Neumann measurements generated by H(λ) for λ in [−λ*, λ*], a strictly smaller set than all von Neumann measurements.
  • Under the pseudo-concavity conjecture (Conjecture 2), the bisection algorithm converges in logarithmic time to the exact accessible information.
  • The closed-form λ* lets one compute the extremal interval directly from the states' purity and overlap, without numerical search.
  • The counterexample with equal marginals pY=pZ shows that even equal output distributions do not restore monotonicity without additional uniformity assumptions.

Reading between the lines

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

  • If the pseudo-concavity conjecture holds, the same restrict-to-extreme-points argument could extend beyond dichotomies to finite-outcome measurements on arbitrary state families.
  • The zero-measure characterization suggests that any future monotonicity-based shortcut must place the distribution exactly on this boundary — a mathematically clean but practically negligible route.
  • A numerical sweep over random qubit dichotomies comparing the bisection algorithm's output with brute-force optimization would provide a fast test of Conjecture 2 before any proof attempt.
  • The explicit counterexample pair could serve as a benchmark for any new claim of monotonicity in information-theoretic quantities.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The paper addresses Shor's conjecture on the accessible information of quantum dichotomies. Its first contribution is a classical-information study of the tradeoff between mutual information and guessing probability for binary joint distributions: Proposition 1 gives a closed-form maximizing distribution under a guessing-probability bound, and Proposition 2 claims to characterize exactly which binary distributions satisfy the monotonicity implication P_X|Y ≥ P_X|Z ⇒ I(X:Y) ≥ I(X:Z), concluding that monotonicity holds only on a zero-measure set. The second contribution is a state-dependent notion of extremality for quantum measurements; Proposition 3 restricts the search for optimal von Neumann measurements for qubit dichotomies to a finite interval λ ∈ [−λ*, λ*], and the paper uses this to tighten Keil's conjecture and propose a bisection algorithm. The paper includes algebraic lemmas, a numerical counterexample in Section II-D, and a code link.

Significance. If correct, the classical result would settle a long-standing disagreement in the literature over monotonicity of mutual information in guessing probability, and the qubit extremality result would provide a tighter, computationally useful characterization of optimal measurements. The paper has clear strengths: the parameterization of binary distributions is convenient, Lemmas 1–3 and Proposition 1 are algebraic and verifiable, the counterexample in Section II-D is concrete, and a reference implementation is provided. However, the central characterization in Proposition 2 is false as stated, which undermines the first main claim until the statement is corrected. The second contribution depends on an external characterization from Ref. [28] that is not re-derived or precisely stated. The paper's significance is therefore conditional on a major revision.

major comments (3)
  1. [Section II-C, Proposition 2] Proposition 2 is false as stated. Equation (4) quantifies over arbitrary binary p_{X,Z}, but the proof immediately imposes the same-marginal constraint Σ_z p(X=0,Z=z)=Σ_y p(X=0,Y=y) in the displayed maximization. A counterexample without that constraint: take p_{X,Y}=[[0.25,0.25],[0,0.5]] and p_{X,Z}=[[0.375,0.25],[0,0.375]]. For p_{X,Y}, Tr p=0.75≥p_X=0=0.5 and P_{X|Y}=0.75=p_Y=0+(1−p_X=0)=0.75, so both conditions of Proposition 2 hold. Yet P_{X|Z}=0.75 while I(X:Y)≈0.311 and I(X:Z)≈0.347, so P_{X|Y}≥P_{X|Z} but I(X:Y)<I(X:Z). Thus the 'if' direction of Proposition 2 is false. The statement must be amended with the same-marginal hypothesis (or an equivalent condition), and the zero-measure conclusion should be re-examined under the corrected statement.
  2. [Section II-C, proof of Proposition 2] The proof as printed is not a proof of the stated theorem. The line 'By setting p_{X,Y}=p(a*,b*,λ*)' is incorrect: the maximizer is a candidate p_{X,Z}, not the given p_{X,Y}. The subsequent inference 'From the third condition we have Tr p_{X,Y}=P_{X|Y}' is also unjustified; third condition refers to I(X:Z)≤I(X:Y), and no relation to the trace is established. Additionally, the argument implicitly uses monotonicity of I(a, λ+a−1, λ) in λ, which is not proved in the paper. The proof needs to be rewritten to prove the corrected statement, with explicit treatment of the marginal constraint and the endpoint maximization in Proposition 1.
  3. [Section III, Proposition 3] The proof of Proposition 3 hinges on the assertion that 'measurements attaining the extremal points of the testing region are those for which λ−≤λ≤λ+ in Eq. (22)', where λ± solve det H(λ)=0. This is a substantive characterization of the Lorenz curve of a qubit dichotomy, imported from Ref. [28] (written by one of the authors), but it is not re-derived or even stated precisely as a lemma in this paper. Since the restriction to λ∈[−λ*,λ*] is the central technical step of the second contribution, the authors should either provide a self-contained proof of this characterization or explicitly state it as an assumption with a precise reference to the exact statement in Ref. [28] and explain the conditions under which it holds. As written, the dependency is not transparent enough for the claimed tightening of Keil's conjecture.
minor comments (3)
  1. [Section II, notation] The symbols δP_{|X|,|Y|} and δP_{|X|,|Y|} are used for both the boundary and the interior of the probability simplex; the notation should be distinguished (e.g., ∂P and int P or similar).
  2. [Section II-C, proof of Proposition 1] The step 'the third equality follows from Property 6 and Property 2 of Lemma 2' is terse. Since the maximization of a convex function over an interval need not occur at the upper endpoint, the authors should explicitly note that the minimum of I(a, λ+a−1, λ) lies at λ=−a and hence the function is increasing for λ≥a, which justifies taking λ=λ*.
  3. [References] Several references (e.g., [33]–[36]) lack full titles and journal/page details. Please complete the bibliography.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: Section II is derived from first principles, and Section III's reliance on Ref. [28] is an independent published characterization, not an assumed target.

full rationale

The information-guessing tradeoff results in Section II are derived from explicit parameterizations, Lemma 2, and convexity/optimization arguments; they do not assume Eq. (4) or Shor's conjecture. Proposition 2 is proved by reducing the question to a maximization solved via Proposition 1 and then identifying the boundary conditions under which a given pXY coincides with that optimizer. Even if the statement of Proposition 2 may be incomplete (the proof imposes a same-marginal constraint not present in the displayed Eq. (4)), that is a correctness/counterexample issue, not circularity: the proof does not use the conclusion as an input. Section III imports the Lorenz-curve characterization that extremal conditional distributions are generated by projectors of H(lambda) from Ref. [28], a prior paper by one of the present authors. This is a self-citation, but it is an independent published mathematical characterization of qubit dichotomies' testing regions, and it does not assume the accessible-information result being sought. The paper also re-derives the purity formula for omega and the explicit expression for lambda* from scratch rather than merely restating them. No parameter is fitted to data, and no quantity that is predicted is defined in terms of the prediction. The self-citation to the implementation [44] is not load-bearing. Overall, the derivation chain does not reduce to its inputs.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper's central claims rest on standard convexity facts, the Lorenz-curve characterization of qubit dichotomies from the authors' prior work, and an unstated monotonicity in Proposition 2. No parameters are fitted; the λ parameterization is an exact variable transformation.

assumptions (3)
  • standard math Mutual information is convex in the conditional distribution P(Y|X) for fixed marginal P(X).
    Used to justify restricting the maximization of accessible information to extreme points of the testing region (Section III, before Prop. 3). Standard result from information theory (Cover-Thomas).
  • domain assumption The testing region / Lorenz curve of a qubit dichotomy has extreme points generated exactly by projectors onto positive/negative parts of H(λ)=λω−(ρ−σ) with λ∈[−λ*,λ*].
    Imported from Ref. [28] (Dall'Arno, prior work by one of the authors). The paper does not prove this characterization in full, but uses it to identify state-dependent extremal measurements in Prop. 3.
  • standard math For a convex function on a compact convex set, the maximum is attained at an extreme point.
    Used in Prop. 3 to restrict from all von Neumann measurements to extremal points of the testing region.

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Cite this review

Pith. "Pith review of On Shor's conjecture on the accessible information of quantum dichotomies." pith.science (2026). https://pith.science/paper/RYFQ2R4L

@misc{pith2026251211233,
  author       = {Pith},
  title        = {Pith review of: On Shor's conjecture on the accessible information of quantum dichotomies},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RYFQ2R4L}},
  note         = {Machine review of arXiv:2512.11233}
}
read the original abstract

Around the turn of the century, Shor formulated his well-known and still-open conjecture stating that the accessible information of any quantum dichotomy, that is the maximum amount of classical information that can be decoded from a binary quantum encoding, is attained by a von Neumann measurement. A quarter of a century later, new developments on the Lorenz curves of quantum dichotomies in the field of quantum majorization and statistical comparison may provide the key to unlock such a longstanding open problem. Here, we first investigate the tradeoff relations between accessible information and guessing probability in the binary case, thus disproving the claimed monotonicity of the former quantity in the latter that, if true, would have settled Shor's problem in the qubit case. Our second result is to provide a state-dependent generalization of extremality for quantum measurements, to characterize state-dependent extremality for qubit dichotomies, and to apply such results to tighten previous results on the accessible information of qubit dichotomies.

Figures

Figures reproduced from arXiv: 2512.11233 by the authors.

Figure 1
Figure 1. Mutual information of binary probability distribution [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

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