Quantum accessible information and classical entropy inequalities

A. S. Holevo; A. V. Utkin

REVIEW 4 major objections 4 minor 16 references

Quantum accessible information and classical entropy inequalities

T0 review · 4 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The quantum-pyramid optimality conjecture reduces to tight classical entropy inequalities.

desk verdict The paper makes a real reduction of the pyramid conjecture to new entropy inequalities, but the multidimensional proofs are sketches and Theorem 3 leans on an unproven two-value hypothesis verified only in a non-reproducible appendix. read the letter →

arxiv 2506.06700 v6 pith:KPLM37GR submitted 2025-06-07 quant-ph cs.ITmath-phmath.ITmath.MP

A. S. Holevo , A. V. Utkin This is my paper

classification quant-phcs.ITmath-phmath.ITmath.MP MSC 81P4594A17

keywords accessibleinformationquantumpyramidentropyinequalityShannonoptimalmeasurementequiangularstatestightlowerboundPOVM

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

What the paper tries to establish is that the long-open problem of computing accessible information for ensembles of equiprobable equiangular pure states (quantum pyramids) can be settled by proving classical entropy inequalities, not by searching over measurements. The key result is a parametric family of tight lower bounds on Shannon entropy, $H(t) \ge \mu_0(p)(\sum_j \sqrt{t_j})^2 - \mu_1(p)$ for $p>(m-1)/m$, with equality at the permutations of $t_1=p$, $t_k=(1-p)/(m-1)$. Using a previously obtained optimality criterion, the authors show that these inequalities are exactly what is needed to certify the globally information-optimal observable conjectured for acute, obtuse, and flat pyramids. If correct, this turns a quantum optimization problem into a finite-dimensional classical inequality and gives closed-form expressions for the accessible information. The proof is complete for the acute case; the obtuse and flat cases rely on a hypothesis about minimizers that is only checked numerically.

What carries the argument

The machinery is the optimality criterion of Theorem 1, which converts the maximization over observables in the definition of accessible information into the minimization of an operator-valued expression, and then into an entropy inequality: a candidate observable is globally optimal exactly when a Hermitian operator $\Lambda_0$ satisfies $\Lambda_0 \le K(\sigma)$ for all states and the candidate states turn the inequality into equality. In the pure-state equiangular case, this reduces to proving a scalar inequality for the Shannon entropy of $t_j=|z_j|^2$, with the coefficients $\mu_0(p),\mu_1(p)$ built from the pyramid's angle parameter $p$. Theorem 2's inequality (23) is the load-bearing inequality: it is the tight lower bound whose tangential points are exactly the states singled out by the conjectured optimal observable. The same criterion, applied to obtuse and flat pyramids, produces inequalities (59), (67), and (71), where the flat case adds the linear constraint $\sum_j z_j=0$.

What would settle it

For some $m\ge7$ and some admissible $a=\tilde\mu_0(p)$, find a point on the unit sphere with three distinct nonzero coordinates whose value of $\tilde F(z)$ in (80) is strictly below every two-value candidate; that would falsify the two-value hypothesis and break the proof of Theorem 3. Alternatively, a direct computational search over $t$ for $m\ge2$ and $p>(m-1)/m$ that violates (23) would refute Theorem 2 and the acute-pyramid corollary.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the optimality criterion from [7] becomes, for quantum pyramids, a statement about the classical Shannon entropy of the probability vector $t_j = |\langle e_j|\psi\rangle|^2$: the optimal measurement exists precisely when a family of tight inequalities like (23), (59), (67), and (71) holds. Theorem 2 states the main inequality: for $m\ge2$ and $p>(m-1)/m$, $-\sum_j t_j\log t_j \ge \mu_0(p)(\sum_j\sqrt{t_j})^2 - \mu_1(p)$, with $\mu_0,\mu_1$ explicit rational-logarithmic functions of $p$ and $m$, and with equality at the stated permutations. The paper argues that this inequality, together with the criterion, proves the global optimality of the conjectured observables and yields formula (52) for the accessible information of acute pyramids, with the obtuse and flat cases covered by inequalities (59), (67), and (71). The authors present a full analytical proof of Theorem 2 and a computer-assisted proof strategy for Theorem 3, conditional on a two-value hypothesis for the minimizer of a sphere optimization problem.

Load-bearing premise

The argument for obtuse and flat pyramids rests on the assumption, verified only numerically, that the global minimizer of the sphere problem (80) has at most two distinct nonzero coordinate values; without a proof of that fact, the derivation of inequalities (59), (67), and (71) does not go through.

Editorial extensions

If this is right

If Theorem 2's inequality (23) holds, the conjectured observable for acute pyramids is globally information-optimal, and the accessible information is given by (52).
The entropy inequality (23) at $p=(m-1)/m$ becomes a lower bound on Shannon entropy in terms of the Bhattacharyya coefficient to the uniform distribution, improving the known total-variation bound (27) near the uniform distribution.
For obtuse pyramids with $m\ge7$, the sharp square-root observable rather than the unambiguous-discrimination observable is tentatively optimal, with the accessible information given by the square-root-measurement expression.
For flat pyramids, inequality (71) gives accessible information $\log(m/2)$ for $m\le6$ and $\frac{m-2}{m}\log(m-1)$ for $m\ge7$.
The optimality criterion provides an approach to proving the pyramid conjectures by independent proof of the entropy inequalities, bypassing direct optimization of POVMs.

Reading between the lines

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

A reader could test Theorem 2 independently of quantum theory: inequality (23) is a purely classical statement, so any counterexample probability vector would refute the acute-pyramid conclusion regardless of the optimality-criterion machinery.
The two-value hypothesis resembles a structure theorem for minimizers of entropy-plus-linear penalties on the sphere; if true for all $m$, it suggests a general classification that may apply to other equiangular ensembles.
The discrete inequalities are described as distant relatives of the log-Sobolev inequality, so a unified proof via rearrangement or mass-transport methods might cover both the discrete and continuous families, though the paper does not develop such a proof.
Because the inequalities are tight at explicit permutations, they could serve as efficient numerical certificates for accessible information in larger equiangular ensembles beyond the pyramids considered here.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Quick take: the two-state case is fully proven and the reduction of the Englert–Rehacek pyramid conjecture to parametric entropy inequalities is genuinely new and worth engaging with. But the multidimensional claim is not yet a proof. Theorem 2 depends on lemmas with steps dismissed as "it is simple to calculate," and Theorem 3 explicitly relies on the hypothesis that minimizers of (80) have at most two distinct nonzero values—a hypothesis checked numerically only for a single parameter value in a DeepSeek-generated appendix, with no code or seeds and using a local search method. The authors themselves call Section 5 "Suggestions for proofs," which is honest but confirming the gap.

What is good: the entropy inequalities (23), (59), (67), (71) appear to be new, and the connection to log-Sobolev relatives is nice. The optimality criterion from [7] is used correctly, and the self-citation is appropriate—it is the external criterion that makes the reduction work. The derivation of the two-state inequality in Proposition 1 is rigorous and gives a clean example of how the method functions.

Where I would push: the acute-pyramid result (Corollary 1) is only as solid as Theorem 2, and Theorem 2's proof has real gaps. The obtuse and flat cases (Theorem 3, Proposition 2, Corollaries 2–3) additionally depend on the two-value hypothesis for m≥7. The numerical appendix does not close that gap: it tests a=10 only, not the full range of tilde-mu0(p), and SLSQP with multiple starts is not a global optimizer. The manuscript itself flags the missing piece, so this is not a case of overclaiming—it is an honest but incomplete attack on a hard problem.

If the lemmas in Section 5.1 can be filled in and the two-value hypothesis either proven or replaced, this would resolve a conjecture open since 2010. As it stands, the paper is a valuable research note with a plausible route, not a final solution. I would send it to a serious referee—ideally someone comfortable with both convex optimization and entropy inequalities—but I would not cite the multidimensional inequalities as established facts in the meantime.

Referee Report

4 major / 4 minor

Summary. The paper applies Holevo's optimality criterion (Theorem 1, from [7]) to two-state ensembles and to equiangular equiprobable quantum pyramids, deriving parametric entropy inequalities: (19) for binary entropy, (23) for acute pyramids, (59) and (67) for obtuse pyramids, and (71) for flat pyramids. The authors claim that, together with Theorem 1, these inequalities imply global optimality of the Englert–Rehacek conjectured observables for quantum pyramids (Corollaries 1–3) and give formulas for accessible information. Proposition 1 for the two-state case is proved completely and yields a tight binary entropy inequality. The multidimensional results are less complete: Theorem 2 is supported only by a proof sketch, and Theorem 3 explicitly relies on an unproven 'two-value hypothesis' for minimizers of the sphere optimization problem (80), with numerical evidence provided in an appendix generated by DeepSeek.

Significance. If the entropy inequalities (23), (59), (67), and (71) are established rigorously, the paper would prove a longstanding conjecture on globally information-optimal measurements for quantum pyramids and would provide new tight classical entropy bounds. The two-state case (Proposition 1) is a clean, fully verified result, and the use of the external optimality criterion from [7] is non-circular and conceptually attractive. The paper also makes the useful methodological point that accessible-information optimality reduces to classical entropy inequalities. However, the multidimensional claims are not yet established: the proof of Theorem 2 is a sketch, and Theorem 3 rests on an explicit hypothesis that is verified only by a non-reproducible local numerical search for a single parameter value. These gaps are load-bearing, so the significance of the paper depends on future completion of the proofs.

major comments (4)

[§5.2, Eq. (78), and the statement 'Our hypothesis is...'] The proof of Theorem 3 and hence Corollaries 2 and 3 relies on the assertion that the minimizer of F̃(z) on the unit sphere has at most two distinct nonzero coordinates. This is explicitly called a hypothesis in Section 5.2 and is not proved. The Lagrange equation (78) can have three distinct real roots, and the proof proceeds by excluding the three-value case only under the stated hypothesis; Lemma 5 covers only two-value candidates. If a three-value minimizer exists for some p in the claimed ranges, inequality (59), Proposition 2, and Corollaries 2–3 are not established. The numerical appendix does not close this gap: it reports results only for a=10, provides no code or random seeds, and SLSQP with multiple starting points is a local search method.
[Appendix, table for m=3–5] The reported verification is internally problematic. For m=3,4,5 the projection onto a two-value structure gives ΔF1 = 0.252, 0.208, 0.122, respectively, which is far above the stated tolerance 1e-8; the table marks these rows 'YES' only because the special vector (1/√2,0,...,0,−1/√2) is within tolerance of the computed minimum. This does not verify the two-value hypothesis for general minimizers, and it leaves open the existence of three-value minima. The algorithm's projection step is a rounding of the computed local minimizer rather than a minimization over the constrained two-value family, so its output cannot certify the hypothesis.
[§5.1, proof of Lemma 4 and Lemma 3] Theorem 2 is presented as a theorem, but its proof depends on Lemmas 3 and 4, whose proofs contain essential steps justified only as 'It is simple to calculate' (e.g., the claim that for β=α one has t*=p and f(p)=0). The reduction of the critical-point check to Lemma 4 also relies on the qualitative behavior of the binary entropy minus a linear-plus-square-root term asserted in Lemma 3. These steps need explicit analytic verification; as written, Theorem 2 is not fully proved.
[§4.4, Table of τo(m) and p(m)] The bifurcation between m≤6 and m≥7, which determines the form of the optimal observable in Theorem 3 and Corollary 2, rests on numerically computed values of τo(m) and p(m), including the negative value reported for m=7. The paper states that p(m) can be expressed via the Lambert W-function, but no proof of the threshold behavior p(m)<(m−1)/m for m≥7 is given. Since Corollary 2 uses this threshold to select between the SRM and the mixture observable, this point should be proved or explicitly labeled as a numerical conjecture.

minor comments (4)

[§4.5] The sentence 'Indeed, the unit vector |ψ⟩ lies in the hyperplane ⟨e0|ψ⟩=0 i equality follows' contains a typo; it should read 'this equality follows'.
[Appendix] The appendix is titled 'Generated by DeepSeek' and is not a conventional reproducible numerical section. The authors should remove this attribution and provide either complete code with seeds and the full set of parameter values, or replace the appendix with a standard numerical study.
[Section 4.1, Eq. (26)] The displayed formula for (26) has an unusual parenthesization in the first line; please check that the coefficient log(m−1)/(m−2) and the following terms are typeset correctly and that the decomposition into the Bhattacharyya form is consistent with (23).
[Reference [10]] Reference [10] includes the title 'Continuity of the relative entropy of resource', which appears unrelated to the cited proof of the orthogonal measurement conjecture for qubit states; please verify the reference details.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the entropy inequalities are proposed as independent classical statements whose analytic proofs do not assume the pyramid conjecture; the optimality criterion is restated and proved in the paper.

full rationale

The paper's load-bearing steps do not reduce to their inputs by construction. The entropy inequalities (23), (59), (67), and (71) are stated as universal lower bounds for Shannon entropy, and the parameters mu0, mu1 are obtained by inserting the conjectured observable into the optimality conditions, but the inequalities themselves are not defined in terms of the target optimality result. Section 5.1 supplies an analytic proof scheme for Theorem 2 via critical-point analysis and lemmas (Lemmas 2-4) that does not invoke the quantum pyramid conjecture. Section 5.2 attempts to prove the obtuse/flat inequalities by reducing the sphere minimization to a two-value hypothesis; the paper explicitly labels this an unproven hypothesis verified only numerically in the DeepSeek-generated appendix for a single parameter value a=10. That is a genuine correctness/completeness gap, but it is not circularity: the hypothesis is an analytical assumption used within a proof, not a restatement of the target inequality. The optimality criterion of Theorem 1 is restated and its proof sketched in the paper, with citation [7] used for technical details; this citation is to the author's prior work but contains a general criterion, not the pyramid conjecture, and it does not assume the paper's conclusions. No equation is fitted to the predicted outcome and then presented as an independent prediction; the accessible-information expressions and the optimality of the conjectured observables are derived from the entropy inequalities plus Theorem 1, rather than being baked into the definitions. Therefore the derivation chain is not circular, despite the acknowledged unproved numerical hypothesis.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted to data; the constants mu0(p), mu1(p), and tau_o(m) are determined by the ensemble geometry and the optimality equations. The paper introduces no new physical entities. The load-bearing unproved inputs are the two-value minimizer hypothesis and the incomplete proofs of Lemmas 3 and 4.

assumptions (4)

domain assumption Optimality criterion (Theorem 1) from Holevo (2022) [7].
The paper's entire method rests on the equivalence between optimality conditions (ii.a)/(ii.b) and the entropy inequality; Section 2 gives only a sketch and the full result is imported from [7].
domain assumption Candidate observable ansatz for quantum pyramids, equations (37) and (54), from Englert-Rehacek [2].
The entropy inequalities are derived by inserting this conjectured optimal observable into the optimality conditions. If the ansatz is not the true optimizer, the resulting inequalities are not relevant to the conjecture.
ad hoc to paper Two-value minimizer hypothesis for the sphere optimization problem (80).
Section 5.2 assumes the global minimizer of the sphere function has at most two distinct nonzero coordinates; this is verified only numerically in the DeepSeek-generated Appendix, with no proof or reproducible code.
ad hoc to paper Lemmas 3 and 4 as used in the proof of Theorem 2.
Theorem 2 reduces to Lemma 4, and the provided proofs of Lemmas 3 and 4 contain gaps, e.g., 'It is simple to calculate that for beta=alpha, we have t*=p'. If these lemmas fail, Theorem 2 is unsupported.

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Cite this review

Pith. "Pith review of Quantum accessible information and classical entropy inequalities." pith.science (2026). https://pith.science/paper/KPLM37GR

@misc{pith2026250606700,
  author       = {Pith},
  title        = {Pith review of: Quantum accessible information and classical entropy inequalities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KPLM37GR}},
  note         = {Machine review of arXiv:2506.06700}
}

read the original abstract

Computing accessible information for an ensemble of quantum states is a basic problem in quantum information theory. We show that the recently obtained optimality criterion (A.S. Holevo, Lobachevskii J. Math., \textbf{43}:7 (2022), 1646-1650), when applied to specific ensembles of states leads to nontrivial tight entropy inequalities that are discrete relatives of the famous log-Sobolev inequality. In this light, the hypothesis of globally information-optimal measurement for an ensemble of equiangular equiprobable states (quantum pyramids) (B.-G. Englert and J. \v{R}eh\'{a}\v{c}ek, J. Mod. Optics \textbf{57 }N3 (2010) 218-226) is reconsidered and the corresponding entropy inequalities are proposed. Via the optimality criterion, this suggests also an approach to the proof of the conjectures concerning globally information-optimal observables for quantum pyramids.

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Reference graph

Works this paper leans on

16 extracted references · 15 canonical work pages

[7]

A. S. Holevo, On Optimization Problem for Positive Operator-Valued Measures, Lobachevskii J. Math.,43:7 (2022), 1646-1650

work page 2022
[1]

E. B. Davies, Information and quantum measurement, IEEE Trans. Inf. Theory24(1978) 596-599

work page 1978
[2]

How well can you know the edge of a quantum pyramid?

B.-G. Englert and J. ˇReh´ aˇ cek, How well can you know the edge of a quantum pyramid, J. Mod. Optics57N3 (2010) 218-226. Arxiv:0905.0510

work page Pith review arXiv 2010
[3]

Gross, Logarithmic Sobolev inequalities, Amer

L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math.97, 1061– 1083 (1975)

work page 1975
[4]

C. W. Helstrom, Quantum Detection and Estimation Theory, (Academic Press, New York 1976)

work page 1976
[5]

A. S. Holevo, Informational aspects of quantum measurement, Probl. Peredachi Inf.9(1973) 31-42; English translation: Probl. Inf. Transm. (USSR)9(1973) 110-118

work page 1973
[6]

A. S. Holevo, Statistical decision theory for quantum systems, J. Multi- variate Anal.,3(1973), 337–394

work page 1973
[8]

A. S. Holevo, An optimization problem concerning noise in quantum measurement channels, Lobachevskii J. Math.,44:6 (2023), 2033-2043

work page 2023

Show all 16 references

[9]

Jozsa, D

R. Jozsa, D. Robb, W. K. Wootters, Lower bound for accessible infor- mation in quantum mechanics, Phys. Rev. A,49:2 (1994), 668-677

1994
[10]

A. Keil, Proof of the orthogonal measurement conjecture for qubit states, Continuity of the relative entropy of resource, IJQI Special Issue Alexander Semenovich Holevo’s 80th Birthday,22N5 2440008 (2024). arXiv:0809.0232 44

2024 arXiv
[11]

A. S. Holevo, D. A. Kronberg, New approaches to eavesdropper infor- mation bounds in quantum cryptography problems, Voprosy kiberbe- zopasnosti, 3(67) (2025), 110-117 (In Russian)

2025
[12]

L. B. Levitin, Optimal quantum measurements for two pure and mixed states, in: Quantum Communications and Measurement, edited by V. P. Belavkin, O. Hirota, and R. L. Hudson, (Plenum Press, New York
[13]

S. M. Moser, Information Theory (Lecture Notes), 6th edition, ETH Z¨ urich and NYCU, 2018

2018
[14]

Sasaki, S

M. Sasaki, S. M. Barnett, R. Jozsa, M. Osaki, and O. Hirota, Acces- sible information and optimal strategies for real symmetrical quantum sources, Phys. Rev.A 59(1999) 3325-3335. arXiv:quant-ph/9812062

1999 arXiv
[15]

P. W. Shor, On the Number of Elements in a POVM Attaining the Accessible Information, arXiv:quant-ph/0009077 (2000)

2000 arXiv
[16]

Suzuki, S

J. Suzuki, S. M. Assad and B.-G. Englert, Accessible information about quantum states: An open optimization problem, in Mathematics of Quantum Computation and Quantum Technology, eds. G. Chen, L. Kauffman and S. J. Lomonaco (Chapman & Hall, 2007), pp. 309-348. 45

2007

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [7]

A. S. Holevo, On Optimization Problem for Positive Operator-Valued Measures, Lobachevskii J. Math.,43:7 (2022), 1646-1650

work page 2022

[2] [1]

E. B. Davies, Information and quantum measurement, IEEE Trans. Inf. Theory24(1978) 596-599

work page 1978

[3] [2]

How well can you know the edge of a quantum pyramid?

B.-G. Englert and J. ˇReh´ aˇ cek, How well can you know the edge of a quantum pyramid, J. Mod. Optics57N3 (2010) 218-226. Arxiv:0905.0510

work page Pith review arXiv 2010

[4] [3]

Gross, Logarithmic Sobolev inequalities, Amer

L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math.97, 1061– 1083 (1975)

work page 1975

[5] [4]

C. W. Helstrom, Quantum Detection and Estimation Theory, (Academic Press, New York 1976)

work page 1976

[6] [5]

A. S. Holevo, Informational aspects of quantum measurement, Probl. Peredachi Inf.9(1973) 31-42; English translation: Probl. Inf. Transm. (USSR)9(1973) 110-118

work page 1973

[7] [6]

A. S. Holevo, Statistical decision theory for quantum systems, J. Multi- variate Anal.,3(1973), 337–394

work page 1973

[8] [8]

A. S. Holevo, An optimization problem concerning noise in quantum measurement channels, Lobachevskii J. Math.,44:6 (2023), 2033-2043

work page 2023