arxiv: 2603.24017 · v2 · submitted 2026-03-25 · 🪐 quant-ph · math-ph· math.FA· math.MP

Recognition: no theorem link

A conjecture on a tight norm inequality in the finite-dimensional l_p

A. S. Holevo , A. V. Utkin

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Pith reviewed 2026-05-15 01:11 UTC · model grok-4.3

classification 🪐 quant-ph math-phmath.FAmath.MP

keywords l_p normtight inequalityRényi entropyquantum channelfinite dimensionconvex functionnorm boundsymmetry

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The pith

A simple tight inequality for l_p norms in any finite dimension d is conjectured to hold.

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

The paper proposes a straightforward but difficult-to-prove inequality that bounds norms of vectors in d-dimensional l_p space. If correct, the bound would deliver sharp constants for comparisons between different p-norms and would simplify related convex optimization tasks. The authors prove the inequality exactly when d equals 3 and supply detailed numerical checks confirming it up to dimension 200. They close by linking the same symmetry ideas to problems of minimizing output Rényi entropy for certain quantum channels.

Core claim

We suggest a tight inequality for norms in d-dimensional l_p space which has simple formulation but appears hard to prove. We give a proof for d=3 and provide a detailed numerical check for d less than or equal to 200 confirming the conjecture. The inequality concerns maximization of a convex function and relies on symmetry properties that also appear in the study of output entropy for quantum channels.

What carries the argument

The conjectured tight l_p-norm inequality, whose bound is achieved at symmetric vectors and supplies the sharp constant for norm comparisons in finite dimensions.

If this is right

The inequality supplies the sharp constant relating different p-norms for any fixed d.
It reduces certain convex maximization problems to checking symmetric cases only.
The same symmetry argument yields the minimum output Rényi entropy for the associated quantum channels.
Numerical verification up to d=200 indicates the bound is saturated by the same symmetric configurations in every dimension.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Analytical proofs for d greater than 3 would immediately give closed-form expressions for entropy extrema that are now obtained only numerically.
The same form of inequality may extend to other Schatten norms or to operators on infinite-dimensional spaces.
Finding the precise vectors that saturate the bound could reveal new symmetry principles useful in quantum information tasks.
A direct computational search for a counterexample in moderate dimensions would settle the conjecture quickly.

Load-bearing premise

The inequality remains exactly true for every dimension d even though it is proven only for d=3 and supported only by numerical checks for larger d.

What would settle it

A single explicit vector in dimension 4 or higher whose l_p norm values violate the conjectured bound.

Figures

Figures reproduced from arXiv: 2603.24017 by A. S. Holevo, A. V. Utkin.

read the original abstract

We suggest a tight inequality for norms in $d$-dimensional space $l_p $ which has simple formulation but appears hard to prove. We give a proof for $d=3$ and provide a detailed numerical check for $d\leq 200$ confirming the conjecture. We conclude with a brief survey of solutions for kin problems which anyhow concern minimization of the output entropy of certain quantum channel and rely upon the symmetry properties of the problem. Key words and phrases: $l_p $-norm, R\'enyi entropy, tight inequality, maximization of a convex function.

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.

Desk Editor's Note private letter to a colleague

Holevo and Utkin state a clean new conjecture on a tight l_p norm inequality, prove it for d=3, and back it with numerics to d=200, but the general case rests on checks that could miss violations.

read the letter

The paper's core is a conjecture giving a simple tight bound on the l_p norm of a vector in d dimensions. They prove the claim outright for d=3 and supply numerical checks that hold through d=200. That combination is the main contribution: a short, explicit statement plus concrete evidence for small and moderate d. The analytic proof for d=3 looks clean and self-contained, and the numerics are described as detailed, which is better than the usual hand-wavy checks one sees in such papers. They also tie the inequality to output entropy minimization for certain quantum channels, which may interest people working on Rényi entropies or channel capacities. The soft spot is exactly what the stress-test note flags. For d>3 the support is numerical only. Exhaustive search over p and over the unit sphere in high dimension is hard; any gap in the grid, any rounding artifact, or any concentration effect at large d could hide a counterexample. The conjecture asserts tightness for every d, so one missed violation would kill it. No asymptotic analysis or reduction to a limiting case is given to close that gap. The paper is therefore best read as a well-supported conjecture rather than a settled theorem. Readers who follow functional analysis or quantum information bounds on finite-dimensional spaces will find it useful for the explicit form and the d=3 proof. It is not yet strong enough to cite as a general fact, but it is coherent and worth referee time. I would send it out for review; the partial result is solid and the open case is clearly stated, so referees can focus on whether the numerics are convincing or whether a general proof is feasible.

Referee Report

1 major / 0 minor

Summary. The manuscript conjectures a tight inequality relating l_p norms of vectors in finite-dimensional spaces R^d, provides a complete analytic proof for the case d=3, supplies detailed numerical verification confirming the inequality for all d up to 200, and surveys related minimization problems for the output Renyi entropy of certain quantum channels that exploit symmetry.

Significance. If the conjecture holds for all d, the inequality would supply a simple, apparently sharp bound on l_p norms with direct utility for bounding Renyi entropies and channel outputs in quantum information. The explicit proof for d=3 and the extensive numerical support constitute concrete strengths; however, the absence of an analytic argument or asymptotic control for general d limits the result's immediate impact.

major comments (1)

[Numerical verification for d ≤ 200] Numerical verification section (d ≤ 200): the checks rely on finite sampling grids over the continuous parameter p and over vectors in R^d; without an explicit error bound, density guarantee, or asymptotic analysis, a violation could remain undetected for some d > 3 or particular p, directly weakening the claim that the inequality is tight for every dimension.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comment on the numerical verification. We address the point below and will make corresponding revisions to improve transparency.

read point-by-point responses

Referee: [Numerical verification for d ≤ 200] Numerical verification section (d ≤ 200): the checks rely on finite sampling grids over the continuous parameter p and over vectors in R^d; without an explicit error bound, density guarantee, or asymptotic analysis, a violation could remain undetected for some d > 3 or particular p, directly weakening the claim that the inequality is tight for every dimension.

Authors: We agree that the numerical checks, although performed on dense grids for p and for vectors on the unit sphere in R^d up to dimension 200, do not come with a rigorous a priori error bound that would exclude all possible violations. In the revised version we will expand the relevant section to report the precise grid resolutions employed (step size in p and angular discretization), the minimal observed ratio of the two sides of the conjectured inequality, and the results of a refinement study for representative values of d showing how the minimal ratio stabilizes. We will also moderate the language to state that the inequality is supported by extensive numerical evidence rather than claiming it is verified for all d. A complete analytic error bound or asymptotic control for general d lies beyond the methods of the present work and would essentially require resolving the full conjecture analytically. revision: partial

Circularity Check

0 steps flagged

No circularity; conjecture with analytic proof for d=3 and separate numerical verification

full rationale

The paper states a conjecture on a tight l_p norm inequality, supplies an explicit proof for the special case d=3, and reports independent numerical checks up to d=200. No equation or claim reduces by construction to its own inputs, no parameter is fitted and then relabeled as a prediction, and no load-bearing step rests on a self-citation whose validity is presupposed by the present work. The numerical evidence is external verification rather than tautological confirmation, and the d=3 proof is self-contained analytic reasoning. The survey of related quantum-channel problems is presented only as context and does not underpin the central conjecture.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract does not specify any free parameters, axioms, or invented entities; the work is a conjecture on an inequality with numerical support.

pith-pipeline@v0.9.0 · 5396 in / 1062 out tokens · 51714 ms · 2026-05-15T01:11:06.132264+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

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

Proof of the Holevo-Utkin conjecture on sharp $\ell_p$ norms for zero-sum vectors
math.CA 2026-05 unverdicted novelty 8.0

Proves that the minimum and maximum of ||x||_p / ||x||_2 over non-zero zero-sum x in R^d equal the stated closed-form expressions for all d ≥ 4.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages · cited by 1 Pith paper · 2 internal anchors

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