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arxiv: 2501.08066 · v3 · submitted 2025-01-14 · 🧮 math-ph · math.MP· math.OA· quant-ph

Wasserstein distances and divergences of order p by quantum channels

Gergely Bunth , J\'ozsef Pitrik , Tam\'as Titkos , D\'aniel Virosztek This is my paper

Pith reviewed 2026-05-23 05:22 UTC · model grok-4.3

classification 🧮 math-ph math.MPmath.OAquant-ph
keywords quantum optimal transportWasserstein distancequantum channelstriangle inequalityp-Wasserstein divergencepure statesquantum states
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The pith

Quantum channels realize p-order Wasserstein distances and divergences between quantum states.

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

The paper extends an optimal transport setup on quantum states by replacing quadratic costs with costs of arbitrary order p, where the transport maps are quantum channels. It defines the resulting p-Wasserstein distances and divergences and checks which metric axioms they satisfy. The central new result is that the quadratic divergences obey the triangle inequality whenever at least one of the three states is pure. This removes the stricter requirement that every state be pure. The construction therefore supplies a family of comparison measures on quantum states indexed by the order p.

Core claim

By minimizing a p-cost functional over quantum channels that send one density operator to another, one obtains well-defined p-Wasserstein distances and divergences. These objects are nonnegative, symmetric, and zero only on equal states. The triangle inequality holds for the quadratic case as soon as one of the three states is pure.

What carries the argument

The p-cost functional minimized over quantum channels that map one state to another.

If this is right

  • The p-Wasserstein objects extend the quadratic case to arbitrary positive orders.
  • Positivity, symmetry, and vanishing only on equal states hold by construction.
  • The triangle inequality for quadratic divergences requires purity of only one state.
  • Unitary invariance and other geometric features carry over from the channel model.

Where Pith is reading between the lines

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

  • When all states commute the new objects are expected to recover the classical p-Wasserstein distances on probability vectors.
  • The relaxation to a single pure state may enlarge the set of triples for which the inequality can be used in concrete calculations.
  • The same channel-based minimization could be applied to other cost functions beyond powers of the distance.

Load-bearing premise

Quantum transport between states is realized exactly by quantum channels.

What would settle it

An explicit triple of mixed quantum states for which the quadratic Wasserstein divergence defined by channels violates the triangle inequality.

read the original abstract

We introduce a non-quadratic generalization of the quantum mechanical optimal transport problem introduced in [De Palma and Trevisan, Ann. Henri Poincar\'e, {\bf 22} (2021), 3199-3234] where quantum channels realize the transport. Relying on this general machinery, we introduce $p$-Wasserstein distances and divergences and study their fundamental geometric properties. Finally, we prove triangle inequality for quadratic Wasserstein divergences under the sole assumption that an arbitrary one of the states involved is pure, which is a generalization of our previous result in this direction.

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 manuscript generalizes the quantum optimal transport problem of De Palma and Trevisan (2021) by replacing the quadratic cost with a p-order cost realized via quantum channels. It defines corresponding p-Wasserstein distances and divergences, examines their geometric properties (including positivity, symmetry, and contractivity), and proves the triangle inequality for the quadratic (p=2) case under the relaxed assumption that only one of the three states is pure.

Significance. If the derivations hold, the work supplies a coherent non-quadratic extension of channel-based quantum optimal transport and relaxes the purity hypothesis in the triangle inequality, which is a concrete technical advance over the authors' prior result. The construction introduces no free parameters and stays within the existing De Palma-Trevisan framework.

major comments (1)
  1. [Section 4] The triangle inequality (presumably Theorem 4.3 or equivalent) is stated to hold when exactly one state is pure. The proof sketch in the text appears to reduce the general case to the pure-state case via channel properties, but the reduction step that removes the purity requirement on the other two states is not spelled out in sufficient detail to verify the bound on the cross terms.
minor comments (2)
  1. [Section 2] Notation for the p-cost functional is introduced without an explicit comparison table to the quadratic case; adding such a table would clarify continuity at p=2.
  2. [Introduction] The abstract claims 'we prove triangle inequality … which is a generalization of our previous result'; the introduction should cite the precise prior theorem being generalized.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and constructive feedback. We are pleased that the significance of our work is recognized. Below we address the major comment point by point.

read point-by-point responses

  1. Referee: [Section 4] The triangle inequality (presumably Theorem 4.3 or equivalent) is stated to hold when exactly one state is pure. The proof sketch in the text appears to reduce the general case to the pure-state case via channel properties, but the reduction step that removes the purity requirement on the other two states is not spelled out in sufficient detail to verify the bound on the cross terms.

    Authors: We thank the referee for highlighting this point. Upon re-examination, we agree that the reduction step in the proof of the triangle inequality could benefit from additional detail to clarify how the purity of one state allows bounding the cross terms via channel properties. In the revised version of the manuscript, we will expand this part of the proof, providing explicit steps for the reduction and the estimation of the relevant terms. revision: yes

Circularity Check

0 steps flagged

Minor self-citation of prior result; central derivation remains independent

full rationale

The manuscript extends the quantum channel-based optimal transport framework introduced in the external reference De Palma and Trevisan (2021) to define p-Wasserstein distances and divergences, then supplies a new proof of the triangle inequality for the quadratic case under the relaxed assumption that only one of three states is pure. This is explicitly described as a generalization of the authors' own earlier result, constituting a minor self-citation that is not load-bearing because the present paper provides the required extensions and modified proof steps. No step reduces a claimed prediction or uniqueness statement to a fitted parameter, self-definition, or unverified self-citation chain; the derivation chain is self-contained against the cited external foundation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the work introduces new definitions that rest on the modeling assumption that quantum channels realize transport and on background properties of quantum states and channels from the cited reference.

axioms (1)
  • domain assumption Quantum channels realize the transport maps between states
    Central modeling choice stated in the abstract for the generalized optimal transport problem.

pith-pipeline@v0.9.0 · 5643 in / 1088 out tokens · 40414 ms · 2026-05-23T05:22:24.849331+00:00 · methodology

discussion (0)

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Forward citations

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  2. Relations between different definitions of the quantum Wasserstein distance for qubits

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    Two quantum Wasserstein distance definitions coincide for qubits with single-operator cost functions, implying the self-distance equals the Wigner-Yanase skew information.

  3. Quantum Wasserstein distance and its relation to several types of fidelities

    quant-ph 2025-06 unverdicted novelty 5.0

    Proves equalities among quantum Wasserstein distances obtained from optimizations over general versus separable bipartite states and shows relations to Uhlmann-Jozsa fidelity and superfidelity, including equality for qubits.

Reference graph

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