{"id":"d7d5aa0e-b05d-4e14-915b-1589529a0e2a","arxiv_id":"2512.11233","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Mutual information is not monotone in guessing probability even for fully binary distributions, and qubit accessible information can be optimized over an explicitly characterized one-parameter family of state-dependent extremal measurements.","lead":"The paper disproves a claimed monotonicity between mutual information and guessing probability for fully binary classical distributions, and characterizes the narrow zero-measure exception set. It also narrows the search space for the optimal measurement in qubit dichotomies, leaving Shor's open conjecture unresolved.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 2 as stated is false: a counterexample meets its two conditions yet violates Eq. (4), because the theorem omits the same-marginal hypothesis its proof assumes.","rationale":"The reader's verdict is CONDITIONAL, and I agree that the paper should not be rejected outright. My concern is more specific than the reader's weakest_assumption: the literal statement of Proposition 2 is false because it omits the same-marginal condition that the proof actually uses. This is load-bearing because Proposition 2 is the paper's first main result, the claimed necessary-and-sufficient characterization of when Eq. (4) holds. The counterexample above is not exotic; it is a simple pair of valid 2x2 joint distributions with equal guessing probability, and it directly violates the implication while satisfying the two stated conditions. The proof's first displayed constraint shows the intended domain is p_{X,Z} with the same p_X as p_{X,Y}; adding that hypothesis excludes the counterexample and is almost certainly the correct fix, which is why the central claim that monotonicity fails except on a zero-measure set may still stand. The reader identified related proof gaps (the omitted monotonicity step and the 'serious typo'), but did not flag the missing hypothesis, so my agreement is partial. I do not see a need to change the reader's CONDITIONAL verdict: the flaw is significant but repairable, and the substantive disproof of monotonicity is likely unaffected. I also note the reader's separate concern about Proposition 3's overclaim for 'any convex objective'; that is a different issue and does not alter this assessment.","tokens_in":13571,"tokens_out":47694,"duration_ms":414783,"concrete_test":"Evaluate the two binary distributions p_{X,Y}=[[0.25,0.25],[0,0.5]] and p_{X,Z}=[[0.375,0.25],[0,0.375]]. Both have P=0.75, and p_{X,Y} satisfies Tr p≥p_{X=0} and P=p_{Y=0}+(1−p_{X=0}), yet I(X:Y)≈0.311 < I(X:Z)≈0.347. This confirms Proposition 2 fails unless Eq. (4) is amended to require p_{X,Z} to have the same X-marginal as p_{X,Y}.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing concern is that Proposition 2 is false as written. Eq. (4) is stated for arbitrary binary p_{X,Z}, with no requirement that p_{X,Z} share the marginal p_X of p_{X,Y}. The proof, however, immediately imposes the constraint Σ_z p(X=0,Z=z)=Σ_y p(X=0,Y=y) in the displayed maximization, so it proves a different statement. The omission is not cosmetic: 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})=0.25+0.5, so both conditions of Proposition 2 hold. Yet P_{X|Z}=0.75 as well, while I(X:Y)=1−0.75·h(2/3)≈0.311 and I(X:Z)=h(0.625)−0.625·h(0.4)≈0.347. Hence P_{X|Y}≥P_{X|Z} but I(X:Y)<I(X:Z), violating Eq. (4). This falsifies the 'if' direction of Proposition 2 unless the same-marginal hypothesis is added to the statement. The zero-measure conclusion about monotonicity may survive, but the closed-form characterization as stated is incorrect.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":13967,"tokens_out":9980,"duration_ms":95799,"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":[{"comment":"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.","section":"Section II-C, Proposition 2"},{"comment":"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.","section":"Section II-C, proof of Proposition 2"},{"comment":"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.","section":"Section III, Proposition 3"}],"minor_comments":[{"comment":"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).","section":"Section II, notation"},{"comment":"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*.","section":"Section II-C, proof of Proposition 1"},{"comment":"Several references (e.g., [33]–[36]) lack full titles and journal/page details. Please complete the bibliography.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The central classical result (Proposition 2) is false as stated; the counterexample is simple and unambiguous. The fix appears to be local—adding the same-marginal hypothesis and rewriting the proof—but until then the paper cannot be accepted. The qubit extremality section is interesting but leans heavily on Ref. [28] without re-deriving the key characterization; the authors should make that dependency explicit. I recommend major revision rather than rejection because the core ideas are valuable and the issues are addressable within the paper's scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the paper has a real contribution — a clean disproof of the claimed monotonicity of mutual information in guessing probability for binary distributions — but its headline characterization, Proposition 2, is false as stated. The stress-test counterexample is correct and lands. Do not reject the whole paper over it; the core result stands and the fix is small.\n\nWhat is actually new and good: Proposition 1 gives a closed-form tradeoff for fixed marginal; Section II-D has an explicit fully-binary counterexample to Eq. (4) even with the same X-marginal, which is the real killer of the monotonicity route. Proposition 3 characterizes the state-dependent extremal measurements for qubit dichotomies and gives an explicit λ* bound. The algebra in Lemmas 1–3 and Proposition 3 checks out. The use of the Lorenz-curve result from Ref. [28] is legitimate — it is published, and they re-derive the specific formulas they need here.\n\nThe soft spot is exactly where the stress-test puts it. Proposition 2 claims to characterize when a binary p_X,Y satisfies Eq. (4) for any binary p_X,Z. The proof actually solves an optimization with the extra constraint that p_X,Z has the same X-marginal as p_X,Y. That hypothesis is not in the statement, and without it the theorem is false. The stress-test example — p_X,Y = [[0.25,0.25],[0,0.5]] and p_X,Z = [[0.375,0.25],[0,0.375]] — satisfies both of Proposition 2's conditions but violates Eq. (4). The proof text is also garbled at the key step: 'By setting p_X,Y = p(a*,b*,λ*)' and 'Tr p_X,Y = P_X|Y' do not follow from anything. The amendment is straightforward: add the same-marginal condition to the statement, and the zero-measure conclusion almost certainly survives. The Section II-D counterexample already shows the monotonicity claim fails independent of Proposition 2.\n\nMinor issue: Proposition 3 says 'any convex objective' without specifying that convexity is in the pair of outcome probabilities. For the mutual information this is fine (fixed prior makes it convex), but the sentence should say so.\n\nWho is this for: people working on accessible information, quantum dichotomies, and the Shor/Keil conjectures. It is a worthwhile paper — the disproof is real, the extremality parameterization is useful, and the conjectures are sharper. But it needs a revised statement and proof of Proposition 2 before it can be trusted as written. Send it to a serious referee.","headline":"Useful disproof of monotonicity, but Proposition 2 is false as stated due to a missing same-marginal hypothesis; still worth refereeing after a fix.","tokens_in":14427,"tokens_out":7745,"would_cite":true,"duration_ms":67869,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81P45","94A17"],"pacs":["03.67.-a"],"model":"deepseek-v4-flash","headline":"More guessable does not mean more informative, even in the binary case","keywords":["accessible information","quantum dichotomy","mutual information","guessing probability","extremal measurements","qubit","testing region","monotonicity"],"falsifier":"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.","tokens_in":13478,"feed_emoji":"🎲","tokens_out":10909,"duration_ms":94334,"temperature":0.7,"pith_summary":"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.","feed_headline":"More guessable does not mean more informative","feed_subtitle":"Even binary channels break the tidy link; the search for optimal qubit measurements tightens to a finite boundary.","key_machinery":"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.","core_discovery":"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","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Guessability doesn't guarantee informativeness for binary quantum codes","Binary quantum states: more guessable ≠ more informative","Qubit measurement optimality pinned to boundary projectors","Tightened accessible-information bounds for qubit dichotomies","Shor's conjecture: guessability and information decoupled in qubits"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Guessability doesn't guarantee informativeness for binary quantum codes","Binary quantum states: more guessable ≠ more informative","Qubit measurement optimality pinned to boundary projectors","Tightened accessible-information bounds for qubit dichotomies","Shor's conjecture: guessability and information decoupled in qubits"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001175,"raw_usage":{"total_tokens":4683,"prompt_tokens":719,"completion_tokens":3964,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":463,"completion_tokens_details":{"reasoning_tokens":3879}},"tokens_in":463,"tokens_out":3964,"duration_ms":29237,"temperature":1.0,"reasoning_tokens":3879,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T16:55:49.103578+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}