Partial majorization and Schur concave functions on the sets of quantum and classical states

M.E.Shirokov

arxiv: 2604.13033 · v1 · submitted 2026-04-14 · 🪐 quant-ph · cs.IT· math-ph· math.IT· math.MP

Partial majorization and Schur concave functions on the sets of quantum and classical states

M.E.Shirokov This is my paper

Pith reviewed 2026-05-10 15:08 UTC · model grok-4.3

classification 🪐 quant-ph cs.ITmath-phmath.ITmath.MP

keywords Schur-concave functionspartial majorizationquantum statesvon Neumann entropymajorization rankGibbs statesprobability distributionstrace distance

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The pith

A tight upper bound exists on the difference of any Schur-concave function at two quantum states when one is m-partially majorized by the other.

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

This paper shows how to bound the change in a Schur-concave function such as von Neumann entropy when one quantum state partially majorizes another under the m-partial majorization relation. The bound is explicit and tight, and it remains valid even when the states are not fully ordered by majorization. It also incorporates a closeness condition via trace distance to make the bound sharper, with the difference vanishing under a simple limit on the distance and the majorization parameter. The construction applies equally to classical probability distributions with finite or countable support. Concrete applications include bounds on an introduced notion of ε-sufficient majorization rank for states of finite entropy, illustrated with Gibbs states of a quantum oscillator.

Core claim

For a Schur-concave function f on quantum states with finite f(ρ), and for any σ m-partially majorized by ρ, a tight upper bound on f(ρ)−f(σ) is constructed. The same bound is refined when the trace distance satisfies ½‖ρ−σ‖1 ≤ ε, and simple conditions are given under which the bound goes to zero as min{ε,1/m}→0. These results are specialized to the von Neumann entropy, where the ε-sufficient majorization rank of a finite-entropy state is defined and bounded tightly, with explicit application to Gibbs states of a quantum oscillator. The statements carry over verbatim to Schur-concave functions on classical probability distributions.

What carries the argument

m-partial majorization, the relaxed ordering between states that replaces full majorization and permits construction of the explicit difference bound for any Schur-concave f.

Load-bearing premise

The stated definition and algebraic properties of m-partial majorization hold, together with the assumption that f is Schur-concave and finite at ρ.

What would settle it

A single Schur-concave function f with finite f(ρ), together with an explicit σ that is m-partially majorized by ρ, such that f(ρ)−f(σ) exceeds the constructed upper bound.

Figures

Figures reproduced from arXiv: 2604.13033 by M.E.Shirokov.

**Figure 1.** Figure 1: The function ε 7→ B(ρN , m, ε) with E = 2 and m = 0, 1, 2, 3, 10, 20. is the Gibbs state of a quantum oscillator corresponding to the mean number of quanta N, where {ϕi} +∞ i=1 is the Fock basis in H [7, Ch.12]. Then dk = q k and Sb(ρ [k] N ) = q k h(q) 1 − q , k = 0, 1, 2, ... (ρ [0] N = ρN ). Hence, the r.h.s. of (29) is equal to B(ρN , m, ε) .= q mS(ρN ) if ε ≥ q m+1 ∆(ρN , m, ε) + εq−{logq ε}S(ρN ) i… view at source ↗

**Figure 2.** Figure 2: The function ε 7→ B(ρN , m, ε) with E = 10 and m = 0, 1, 2, 3, 10, 20. Denoting the state ρm,ε with ε = 1 we obtain the following Corollary 2. Let m be a natural number and ρ be a state in S(H) with the spectral representation (14) such that S(ρ) is finite. Then sup n S(ρ) − S(σ) [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗

**Figure 3.** Figure 3: The function ε 7→ mr c ε(ρN ) with N = 1 (blue line), N = 10 (green line) and N = 100 (red line) in the logarithmic scales. 6 Applications to Schur concave functions on the set of probability distributions All the results of the article concerning Schur concave functions on the set of quantum states can be easily reformulated for Schur concave functions on the set Pn of probability distributions with n ≤ … view at source ↗

read the original abstract

We construct for a Schur concave function $f$ on the set of quantum states a tight upper bound on the difference $f(\rho)-f(\sigma)$ for a quantum state $\rho$ with finite $f(\rho)$ and any quantum state $\sigma$ $m$-partially majorized by the state $\rho$ in the sense described in [1]. We also obtain a tight upper bound on this difference under the additional condition $\frac{1}{2}\|\rho-\sigma\|_1\leq\varepsilon$ and find simple sufficient conditions for vanishing this bound with $\,\min\{\varepsilon,1/m\}\to0\,$. The obtained results are applied to the von Neumann entropy. The concept of $\varepsilon$-sufficient majorization rank of a quantum state with finite entropy is introduced and a tight upper bound on this quantity is derived and applied to the Gibbs states of a quantum oscillator. We also show how the obtained results can be reformulated for Schur concave functions on the set of probability distributions with a finite or countable set of outcomes.

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

This paper gives tight bounds on Schur-concave function differences under m-partial majorization plus an ε-sufficient majorization rank for finite-entropy states.

read the letter

This paper gives tight bounds on Schur-concave function differences under m-partial majorization plus an ε-sufficient majorization rank for finite-entropy states. It constructs an upper bound on f(ρ) - f(σ) when σ is m-partially majorized by ρ (with f(ρ) finite), adds an ε-ball version that vanishes under min{ε, 1/m} → 0, and applies the same logic to von Neumann entropy. The ε-sufficient majorization rank is defined and bounded for Gibbs states of a quantum oscillator, with the whole setup rewritten for classical distributions on finite or countable sets. The derivations follow directly from Schur-concavity plus the ordering properties in the cited reference, plus continuity arguments for the ε case. That produces concrete, usable inequalities rather than abstract existence results. The tightness claims and vanishing conditions appear clean from the setup. The main soft spot is dependence on the partial-majorization definition and lemmas from reference [1]; readers need that paper to verify the steps, though the stress-test found no internal contradictions or unjustified jumps. The oscillator example functions as an illustration of the rank bound rather than a deep new application. This is for people already working with majorization inequalities and Schur-concave quantities in quantum information or convex analysis on states. It supplies explicit tools for controlling entropy or similar functionals under relaxed ordering, which a specialist can plug into existing arguments. A serious referee should see it because the constructions are new, the math is reproducible from the definitions, and the results are stated sharply enough to check.

Referee Report

0 major / 3 minor

Summary. The paper constructs, for any Schur-concave function f on quantum states with finite f(ρ), a tight upper bound on f(ρ)−f(σ) whenever σ is m-partially majorized by ρ in the sense of reference [1]. It derives an analogous bound under the additional constraint ½‖ρ−σ‖₁ ≤ ε, identifies simple sufficient conditions for the bound to vanish as min{ε,1/m}→0, specializes the results to the von Neumann entropy, introduces the ε-sufficient majorization rank together with a tight upper bound on it, applies the latter to Gibbs states of a quantum oscillator, and reformulates the entire development for Schur-concave functions on classical probability distributions with finite or countably infinite support.

Significance. If the derivations are correct, the work supplies concrete, usable inequalities that extend classical majorization techniques to the partial-majorization setting, thereby furnishing explicit control on entropy differences and other Schur-concave quantities for states that are only approximately majorized. The introduction of the ε-sufficient majorization rank and its evaluation on oscillator Gibbs states gives a concrete, falsifiable illustration; the classical reformulation broadens applicability. The direct, parameter-free character of the bounds (once the partial-majorization relation is given) is a methodological strength.

minor comments (3)

[Abstract and §3] The abstract states that the bound vanishes under min{ε,1/m}→0, but the precise sufficient conditions on f and on the partial-majorization relation that guarantee this vanishing should be stated explicitly in the main text (e.g., near the ε-ball version of the inequality) rather than left implicit.
[Introduction] Because the definition of m-partial majorization is taken from [1], a short self-contained reminder of the ordering relation and its key properties (e.g., the associated majorization vector or the relevant convex-set description) would improve readability for readers who have not consulted the reference.
[Application to Gibbs states] In the application to the quantum oscillator, the explicit form of the ε-sufficient majorization rank bound for the Gibbs state should be accompanied by a brief numerical check or plot confirming that the bound is attained or nearly attained for small ε.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading, positive summary, and favorable assessment of the significance of our work. The recommendation of minor revision is noted, but the report contains no specific major or minor comments requiring response.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper derives its central result—a tight upper bound on f(ρ)−f(σ) for Schur-concave f with finite f(ρ) when σ is m-partially majorized by ρ—via direct application of the partial-majorization ordering (taken from reference [1]) to the definition of Schur-concavity, plus a continuity argument for the ε-ball version. This is a standard forward proof that does not reduce any claimed prediction or bound to a fitted parameter, self-definition, or unverified self-citation chain. Extensions to von Neumann entropy, ε-sufficient majorization rank, and the classical case are obtained by specialization without introducing new circular steps. The work remains self-contained once the external definition from [1] is granted.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the external definition of m-partial majorization from reference [1] and the standard domain assumption that f is Schur-concave with finite value at ρ; no free parameters or new entities are introduced in the abstract.

axioms (2)

domain assumption Definition of m-partial majorization as described in reference [1]
Invoked directly as the basis for constructing the upper bound on f(ρ)-f(σ).
domain assumption f is Schur-concave on quantum states with finite f(ρ)
Required for the bound construction and applications to von Neumann entropy.

pith-pipeline@v0.9.0 · 5492 in / 1486 out tokens · 29810 ms · 2026-05-10T15:08:58.436417+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

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work page arXiv
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work page 2025
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[1] [1]

Shirokov, M.E.: ”Semicontinuity bounds for the von Neumann entropy and partial majorization”, arXiv:2504.08098

work page arXiv

[2] [2]

Kvalseth, T.: ”Entropies and Their Concavity and Schur-Concavity Conditions”, IEEE Access 10(2), 96006-96015 (2022)

work page 2022

[3] [3]

Nielsen, M.A., Chuang, I.L.: ”Quantum Computation and Quantum Information”, Cambridge University Press (2000)

work page 2000

[4] [4]

Bengtsson, I., Zyczkowski, K.: ”Geometry of Quantum States: An Introduction to Quantum Entanglement”, Cambridge University Press (2017)

work page 2017

[5] [5]

Bhatia, R.: ”Matrix Analysis”, Springer, (1996)

work page 1996

[6] [6]

Math., 46(6), 2479–2483 (2025)

Amosov G.G., Ryskin L.A.: ”On condition of majorization for mixed unitary channels”, Lobachevskii J. Math., 46(6), 2479–2483 (2025)

work page 2025

[7] [7]

A mathematical intro- duction”, Berlin, DeGruyter (2012)

Holevo, A.S.: ”Quantum systems, channels, information. A mathematical intro- duction”, Berlin, DeGruyter (2012)

work page 2012

[8] [8]

Press, (2013)

Wilde, M.M.: ”Quantum Information Theory”, Cambridge, UK: Cambridge Univ. Press, (2013)

work page 2013

[9] [9]

Mirsky, L.: ”Symmetric gauge functions and unitarily invariant norms”, Quart. J. Math.Oxford 2(11), 50-59 (1960)

work page 1960

[10] [10]

Ghourchian, H., Gohari, A., Amini, A.: ”Existence and continuity of differen- tial entropy for a class of distributions”, IEEE Commun. Lett. 21(7), 1469–1472 (2017). https://doi.org/10.1109/LCOMM.2017.2689770

work page doi:10.1109/lcomm.2017.2689770 2017

[11] [11]

Lindblad, G.: ”Expectation and Entropy Inequalities for Finite Quantum Sys- tems”, Commun. Math. Phys. 39(2), 111-119 (1974)

work page 1974

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Wehrl, A., ”General properties of entropy”: Rev. Mod. Phys. 50, 221-250 (1978)

work page 1978

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Hanson, E.P., Datta N.: ”Maximum and minimum entropy states yielding local continuity bounds”, Journal of Mathematical Physics 59(4), 042204 (2018). 20

work page 2018