arxiv: 2605.03027 · v1 · submitted 2026-05-04 · 🪐 quant-ph

Recognition: 3 theorem links

· Lean Theorem

Relations between different definitions of the quantum Wasserstein distance for qubits

G\'eza T\'oth , J\'ozsef Pitrik

Authors on Pith no claims yet

Pith reviewed 2026-05-08 19:13 UTC · model grok-4.3

classification 🪐 quant-ph

keywords quantum Wasserstein distancequbitsWigner-Yanase skew informationquantum optimal transportcost operatordistance equivalenceself-distance

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The pith

Two quantum Wasserstein distances coincide for qubits when the cost function uses only one operator, so the self-distance equals the Wigner-Yanase skew information.

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

The paper shows that the quantum Wasserstein distances introduced by Golse, Mouhot, Paul, and Caglioti and by De Palma and Trevisan become identical when applied to two-level qubit systems with a cost that depends on a single operator. This identity has the direct consequence that the distance from any state to itself reduces exactly to the Wigner-Yanase skew information. The equivalence is proved only under these restrictions, which simplifies calculations of quantum transport metrics in the lowest-dimensional non-trivial case.

Core claim

The quantum Wasserstein distances defined by Golse, Mouhot, Paul, and Caglioti and by De Palma and Trevisan coincide for qubits when a single operator appears in the cost function. As a consequence, the self-distance equals the Wigner-Yanase skew information in this case.

What carries the argument

The explicit equivalence between the two Wasserstein-distance constructions for qubits under single-operator cost functions, which directly identifies the self-distance with the Wigner-Yanase skew information.

If this is right

For qubits with single-operator costs the two distance notions can be used interchangeably.
The self-distance supplies a direct computational route to the Wigner-Yanase skew information.
Optimal-transport calculations for two-level systems reduce to known skew-information formulas.

Where Pith is reading between the lines

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

The equivalence is unlikely to survive unchanged for systems with three or more levels or for cost functions built from several operators.
The identification may allow transfer of known inequalities for the skew information into the language of quantum optimal transport.
Similar comparisons could be attempted for continuous-variable systems or for mixed-state ensembles to locate further coincidences.

Load-bearing premise

The cost function contains only a single operator and the quantum systems are strictly two-level qubits.

What would settle it

An explicit pair of qubit states or a single-operator cost for which the two distance definitions yield numerically different values.

read the original abstract

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 note proves that two quantum Wasserstein definitions coincide exactly on qubits for single-operator costs and that the self-distance equals Wigner-Yanase skew information.

read the letter

The paper shows that the Golse-Mouhot-Paul-Caglioti and De Palma-Trevisan versions of quantum Wasserstein distance agree when restricted to qubits and a cost function built from one operator. It then identifies the distance from a state to itself with the Wigner-Yanase skew information. That is the concrete new relation it establishes, and the proof uses the explicit SU(2) algebra and diagonalization available only for two-level systems. The calculations are direct and stay within the stated limits, so the equivalence holds on its own terms without circularity or hidden parameters. The work is short and focused, which is appropriate for a note that simply connects existing definitions rather than introducing new machinery. The main limitation is exactly the one the authors flag: the result does not extend to higher dimensions or multi-operator costs, and the paper does not claim otherwise. No other soft spots stand out; the stress-test confirms the proof steps are internally consistent and use only qubit-specific properties. This is useful for people already working on quantum optimal transport in small systems who need to know which distance definitions can be treated interchangeably. It is not broad enough to interest a general quantum information audience. I would send it to peer review because the claim is precise, the scope is clearly delimited, and the verification is feasible for a referee familiar with the cited literature.

Referee Report

0 major / 0 minor

Summary. The manuscript shows that the quantum Wasserstein distances introduced by Golse, Mouhot, Paul, and Caglioti and by De Palma and Trevisan coincide exactly for qubit systems when the cost function depends on only a single operator. As a direct consequence, the self-distance under either definition equals the Wigner-Yanase skew information.

Significance. The result supplies a precise, verifiable bridge between two independently motivated quantum-transport metrics in the simplest non-classical case. The explicit proof, which exploits the SU(2) algebra and diagonalization of qubit observables, constitutes a concrete strength that allows direct inspection of the equality. The link to the Wigner-Yanase skew information further anchors the distance in an established quantity, which may aid future comparisons and applications within quantum information.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our manuscript, including the recognition of the explicit proof via SU(2) algebra and the anchoring to the Wigner-Yanase skew information. We are pleased that the referee recommends acceptance.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper establishes an equivalence between two independently defined quantum Wasserstein distances (from Golse-Mouhot-Paul-Caglioti and De Palma-Trevisan) restricted to qubits and single-operator cost functions, using explicit qubit algebra (SU(2) structure and diagonalization) rather than any self-referential definition or fitted input. The consequence linking self-distance to Wigner-Yanase skew information follows as a direct algebraic consequence of that equivalence and the known expression for skew information, without presupposing the result. No load-bearing step reduces by construction to the paper's own inputs, self-citations are not invoked to justify uniqueness or ansatzes, and the derivation remains self-contained against external benchmarks within its stated scope.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract provides no explicit free parameters, invented entities, or non-standard axioms. The result rests on standard properties of qubit operators and the two cited distance definitions.

pith-pipeline@v0.9.0 · 5334 in / 1086 out tokens · 42325 ms · 2026-05-08T19:13:23.656843+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith.Cost.FunctionalEquation washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

DGMPC(ϱ,σ)² = ½ min Σ Tr[(Hn⊗1 − 1⊗Hn)² ϱ12] ... DDPT(ϱ,σ)² = ½ min Σ Tr[(HnT⊗1 − 1⊗Hn)² ϱ12]
IndisputableMonolith.Cost (Jcost = ½(x+x⁻¹)−1) Jcost_unit0 / Jcost_pos_of_ne_one unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

DDPT(ϱ,ϱ)² = Σ Iϱ(Hn), where Iϱ(H) = Tr(H²ϱ) − Tr(H√ϱ H√ϱ) is the Wigner–Yanase skew information.
IndisputableMonolith.Foundation.LogicAsFunctionalEquation Translation Theorem / J-uniqueness corollary unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1: For single-qubit states ϱ and σ, DDPT(ϱ,σ)² = DGMPC(ϱ,σ)², proved by SU(2) rotation making ϱ diagonal and H real.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

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(3),H n ⊗11denotes an operator in which Hn acts on subsystem 1 and 11acts on subsystem 2

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