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arxiv: 2507.07520 · v3 · submitted 2025-07-10 · 🪐 quant-ph · cs.IT· math.IT

Conditions for Large-Sample Majorization of Pairs of Flat States in Terms of α-z Relative Entropies

Frits Verhagen , Marco Tomamichel , Erkka Haapasalo This is my paper

Pith reviewed 2026-05-19 06:01 UTC · model grok-4.3

classification 🪐 quant-ph cs.ITmath.IT
keywords relative majorizationflat statesalpha-z relative entropyquantum resource theorylarge-sample transformationscatalytic conversionspreordered semirings
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The pith

The α-z relative entropies for α less than 1 fully determine when one pair of flat quantum states can be converted into another by large-sample or catalytic majorization.

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

The paper shows that large-sample or catalytic relative majorization between pairs of flat states holds exactly when the α-z relative entropies with α below 1 are ordered in the same way. This supplies the first operational reading of these entropies as the quantities that govern concrete state-conversion tasks. A reader cares because the result turns an abstract family of distinguishability measures into a practical test for whether one pair of states can be transformed into another and at what rate. The proof relies on real-algebraic methods with preordered semirings, monotone homomorphisms, and derivations, and it treats the parameters α and z as fully independent.

Core claim

Transformations between pairs of flat states (and certain generalizations) via large-sample or catalytic relative majorization exist if and only if all α-z relative entropies for α less than 1 are ordered between the two pairs; the same ordering also determines the optimal conversion rate.

What carries the argument

The α-z relative entropies (with α < 1) acting as the complete set of monotone invariants that characterize relative majorization orderings on flat-state pairs.

If this is right

  • The parameters α and z remain independent in the resulting conditions.
  • The ordering directly supplies the optimal asymptotic rate for converting one flat-state pair into the other.
  • The same characterization covers certain natural generalizations of flat states.
  • The algebraic techniques yield necessary and sufficient conditions rather than merely sufficient ones.

Where Pith is reading between the lines

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

  • The result suggests that similar algebraic invariants might classify conversions for non-flat states if the semiring structure can be extended.
  • It connects the abstract family of α-z entropies to resource-conversion problems in quantum thermodynamics and information processing.
  • Numerical checks on small-dimensional flat states could confirm the rate formula before analytic proofs are attempted for broader classes.

Load-bearing premise

The states are exactly flat or admit simple generalizations, and the real-algebraic structure of preordered semirings with monotone homomorphisms applies without extra restrictions.

What would settle it

A pair of concrete flat states (for example, two distinct pairs of uniform distributions on different numbers of qubits) for which the α-z relative entropies with α < 1 are ordered but no large-sample majorization transformation exists at the predicted rate.

Figures

Figures reproduced from arXiv: 2507.07520 by Erkka Haapasalo, Frits Verhagen, Marco Tomamichel.

Figure 1
Figure 1. Figure 1: A depiction of the family of all relative entropies given by (11) and (12). For α ∈ (0, 1) and z ≥ max{α, 1 − α}, Dˆ α,z equals the α-z relative en￾tropies given by (8), up to a prefactor. With this different prefactor, the α-z divergences can be extended easily to the segments α = 0 or α = 1, and z ≥ 1. The additional relative entropy Dˆ T = limz→∞ Dˆ α,z, in￾dependent of α, turns the set of all relative … view at source ↗
read the original abstract

We offer the first operational interpretation of the $\alpha$-z relative entropies, a measure of distinguishability between two quantum states introduced by Jak\v{s}i\'c et al. and Audenaert and Datta. We show that these relative entropies appear when formulating conditions for large-sample or catalytic relative majorization of pairs of flat states and certain generalizations of them. Indeed, we show that such transformations exist if and only if all the $\alpha$-z relative entropies for $\alpha$<1 of the two pairs are ordered. In this setting, the $\alpha$ and z parameters are truly independent from each other. These results also yield an expression for the optimal rate of converting one flat state pair into another. Our methods use real-algebraic techniques involving preordered semirings and certain monotone homomorphisms and derivations on them.

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.

Referee Report

1 major / 2 minor

Summary. The paper claims to give the first operational interpretation of the α-z relative entropies by showing that, for pairs of flat quantum states and certain generalizations, large-sample or catalytic relative majorization exists if and only if the α-z relative entropies with α<1 are ordered between the two pairs. The result also supplies the optimal conversion rate. The proof relies on real-algebraic methods that equip the set of flat-state pairs with the structure of a preordered semiring and then characterize the allowed transformations via monotone homomorphisms and derivations on that semiring.

Significance. If the central iff statement holds, the work supplies a concrete operational meaning to the α-z family in a resource-theoretic setting and demonstrates that α and z remain independent parameters. The algebraic approach yields a clean, parameter-free characterization and an explicit rate formula; these are genuine strengths that could influence how distinguishability measures are used in quantum thermodynamics and majorization-based resource theories.

major comments (1)
  1. [Main theorem and semiring construction (around the proof of sufficiency)] The sufficiency direction of the main iff claim (the existence of the large-sample map whenever the α-z ordering holds) rests on the assertion that every monotone homomorphism of the preordered semiring of flat-state pairs is realized by some α-z relative entropy with α<1. If additional order-preserving functionals exist that are not captured by this family, the ordering condition would be necessary but not sufficient. The manuscript should supply an explicit argument that the derivations and homomorphisms constructed in the semiring framework exhaust all possible invariants for flat states.
minor comments (2)
  1. [Introduction] The phrase 'certain generalizations of them' appears in the abstract and introduction but is not illustrated with a concrete example until later; adding a short motivating example immediately after the statement of the main result would improve readability.
  2. [Preliminaries] Notation for the pair of flat states (ρ,σ) and the associated semiring operations could be summarized in a small table or displayed equation early in the preliminaries to help readers track the algebraic structure.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive evaluation of the manuscript and for identifying a point that can be clarified to strengthen the presentation of the sufficiency direction. We address the major comment in detail below and will incorporate revisions accordingly.

read point-by-point responses

  1. Referee: [Main theorem and semiring construction (around the proof of sufficiency)] The sufficiency direction of the main iff claim (the existence of the large-sample map whenever the α-z ordering holds) rests on the assertion that every monotone homomorphism of the preordered semiring of flat-state pairs is realized by some α-z relative entropy with α<1. If additional order-preserving functionals exist that are not captured by this family, the ordering condition would be necessary but not sufficient. The manuscript should supply an explicit argument that the derivations and homomorphisms constructed in the semiring framework exhaust all possible invariants for flat states.

    Authors: We appreciate the referee's careful scrutiny of the sufficiency proof. The manuscript establishes that the monotone homomorphisms on the preordered semiring of flat-state pairs are realized exactly by the α-z relative entropies for α<1 through the algebraic characterization in Theorem 3.5 and the supporting lemmas on derivations (Section 4). These results rely on the real-algebraic structure to show that any order-preserving functional must arise from this parameterized family, with α and z remaining independent. To make the exhaustion of all invariants fully explicit as requested, we will add a dedicated clarifying paragraph immediately following the statement of the main theorem, together with a short appendix subsection that recalls the relevant semiring homomorphism classification and confirms no additional functionals exist for this specific preorder. This revision will render the sufficiency direction self-contained without changing any theorems or proofs. revision: yes

Circularity Check

0 steps flagged

No significant circularity; algebraic framework yields independent characterization

full rationale

The paper derives the iff condition for large-sample/catalytic majorization of flat-state pairs from the ordering of α-z relative entropies (α<1) by applying general real-algebraic tools: preordered semirings, monotone homomorphisms, and derivations. These structures are not defined in terms of the majorization relation or the target ordering; they are standard mathematical objects used to characterize allowed transformations. The α-z relative entropies themselves are cited from independent prior works (Jakšić et al., Audenaert and Datta) and are not introduced or fitted within this manuscript. Necessity follows directly from monotonicity of the homomorphisms; sufficiency follows from the existence of derivations that realize the maps when the ordering holds. No self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citations appear. The derivation is self-contained against external algebraic benchmarks and does not reduce the claimed result to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on standard mathematical background from algebra and quantum information; no free parameters or new postulated entities are introduced in the abstract.

axioms (1)
  • standard math Properties of preordered semirings and existence of monotone homomorphisms and derivations
    Invoked in the methods section to establish the equivalence.

pith-pipeline@v0.9.0 · 5693 in / 1158 out tokens · 33366 ms · 2026-05-19T06:01:06.780956+00:00 · methodology

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