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arxiv: 2510.26326 · v2 · submitted 2025-10-30 · 🧮 math-ph · math.MP· math.OA· quant-ph

Strong Kantorovich duality for quantum optimal transport with generic cost and optimal couplings on quantum bits

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

Pith reviewed 2026-05-18 03:44 UTC · model grok-4.3

classification 🧮 math-ph math.MPmath.OAquant-ph
keywords quantum optimal transportKantorovich dualityquantum Wasserstein divergencequantum bitstriangle inequalityoptimal couplingsquantum channels
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The pith

Kantorovich duality holds for linearized quantum optimal transport realized by quantum channels

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

The paper proves Kantorovich duality for a linearized version of a non-quadratic quantum optimal transport problem in which quantum channels realize the transport maps. This duality is then applied to quantum bits with distinguished cost operators and under restrictions on the states to obtain explicit optimal solutions for both the primal and dual problems. The same restrictions allow an analytical proof that the square of the induced quantum Wasserstein divergence satisfies the triangle inequality. A sympathetic reader would care because the results supply concrete analytical tools for distances between quantum states when standard quadratic costs are replaced by generic ones.

Core claim

We prove Kantorovich duality for the linearized quantum optimal transport problem with generic cost, where quantum channels realize the transport. As an application, we determine optimal solutions of both the primal and the dual problem using this duality in the case of quantum bits and distinguished cost operators, with certain restrictions on the states involved. Finally, keeping the same restrictions regarding the states involved, we use this information on optimal solutions to give an analytical proof of the triangle inequality even for the square of the induced quantum Wasserstein divergences.

What carries the argument

The linearized formulation of the non-quadratic quantum optimal transport problem with quantum-channel couplings, for which strong Kantorovich duality is established and then used to extract optimal solutions

If this is right

  • Explicit optimal primal and dual solutions become available for qubit states under the given restrictions and distinguished costs.
  • The square of the induced quantum Wasserstein divergence satisfies the triangle inequality under the same state restrictions.
  • Analytical verification of metric properties is possible for this channel-based quantum transport setting.

Where Pith is reading between the lines

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

  • The duality and triangle inequality might extend to higher-dimensional systems if analogous state restrictions can be identified.
  • Numerical checks on small quantum systems outside the restricted class could test whether the results survive relaxation of the assumptions.
  • The linearized channel formulation may suggest approximation schemes that connect to classical optimal transport computations.

Load-bearing premise

The restrictions placed on the states involved are sufficient to determine explicit optimal solutions for both primal and dual problems and to carry the triangle inequality proof for the squared divergence.

What would settle it

A concrete counterexample consisting of two qubit states satisfying the stated restrictions, a distinguished cost operator, and a quantum channel for which either the duality equality fails or the squared Wasserstein divergence violates the triangle inequality would disprove the central results.

read the original abstract

We prove Kantorovich duality for a linearized version of a recently proposed non-quadratic quantum optimal transport problem, where quantum channels realize the transport. As an application, we determine optimal solutions of both the primal and the dual problem using this duality in the case of quantum bits and distinguished cost operators, with certain restrictions on the states involved. Finally, keeping the same restrictions regarding the states involved, we use this information on optimal solutions to give an analytical proof of the triangle inequality even for the square of the induced quantum Wasserstein divergences.

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 / 3 minor

Summary. The manuscript proves Kantorovich duality for a linearized version of a non-quadratic quantum optimal transport problem in which quantum channels realize the transport maps. As an application, explicit optimal solutions for both the primal and dual problems are derived in the qubit case for distinguished cost operators under stated restrictions on the input states. These optima are then used to give an analytical proof of the triangle inequality for the square of the induced quantum Wasserstein divergence, again under the same state restrictions.

Significance. If the duality and the subsequent derivations hold, the work supplies a concrete duality theorem for a linearized quantum OT problem with generic costs and yields explicit optima on qubits together with a direct analytical verification of the triangle inequality for the squared divergence. The analytical (non-numerical) character of the triangle-inequality argument is a clear strength. No machine-checked proofs or parameter-free derivations are claimed, but the explicit construction of optima and their use in the inequality proof constitute a useful contribution to the quantum optimal transport literature.

major comments (1)
  1. [Section 2 (duality theorem) and Section 4 (triangle inequality)] The duality theorem is stated for generic costs, yet the explicit optima and the triangle-inequality argument are derived only after imposing restrictions on the states. The manuscript should clarify, with a precise statement (e.g., in the paragraph following the duality theorem), whether the duality itself requires any of these restrictions or holds without them; otherwise the scope of the central claim remains ambiguous.
minor comments (3)
  1. [Section 3] The precise definition of the 'distinguished cost operators' and the exact form of the state restrictions should be collected in a single displayed assumption box or numbered list for easy reference.
  2. [Introduction] Notation for the quantum channels realizing transport and for the linearized cost functional is introduced gradually; a short table summarizing all symbols at the end of the introduction would improve readability.
  3. [Abstract] The abstract claims a 'strong' Kantorovich duality; a brief sentence relating this terminology to the classical or quantum literature would help readers situate the result.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation and for identifying the need to clarify the scope of the duality result. We address the single major comment below and will incorporate the requested clarification.

read point-by-point responses

  1. Referee: [Section 2 (duality theorem) and Section 4 (triangle inequality)] The duality theorem is stated for generic costs, yet the explicit optima and the triangle-inequality argument are derived only after imposing restrictions on the states. The manuscript should clarify, with a precise statement (e.g., in the paragraph following the duality theorem), whether the duality itself requires any of these restrictions or holds without them; otherwise the scope of the central claim remains ambiguous.

    Authors: The Kantorovich duality (Theorem 2.1) is formulated and proved for arbitrary cost operators on finite-dimensional quantum systems; its statement and proof impose no restrictions on the input states. The restrictions on the states appear only from Section 3 onward, where we specialize to qubits and distinguished costs in order to obtain explicit primal and dual optima. The same restrictions are retained in Section 4 solely to obtain an analytical proof of the triangle inequality for the squared divergence. We agree that the distinction should be stated explicitly. In the revised version we will add, immediately after the statement of Theorem 2.1, a short paragraph clarifying that the duality holds for generic costs and without state restrictions, while the subsequent explicit constructions and the triangle-inequality argument are subject to the stated assumptions on the states. revision: yes

Circularity Check

0 steps flagged

No circularity: direct analytical proofs of duality and triangle inequality under explicitly maintained restrictions

full rationale

The paper states it proves Kantorovich duality for a linearized version of a prior quantum OT problem, then applies the duality to obtain explicit primal/dual optima for qubits with distinguished costs under stated state restrictions, and finally uses those optima to prove the triangle inequality for the squared divergence while keeping the identical restrictions. No quoted step shows a result reducing by construction to a fitted input, a self-citation chain, or a renamed ansatz; the derivation chain consists of stated proofs and applications that remain self-contained within the given restrictions without invoking unverified load-bearing prior results as substitutes for the new arguments.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard mathematical framework of quantum states, channels, and convex optimization; no free parameters, invented entities, or ad-hoc axioms are mentioned in the abstract.

axioms (2)
  • domain assumption Quantum states are density operators and transport is realized by completely positive trace-preserving maps
    Invoked throughout the abstract as the setting for the optimal transport problem.
  • domain assumption The cost operator is a fixed self-adjoint operator on the tensor product space
    Used to define the linearized transport cost.

pith-pipeline@v0.9.0 · 5636 in / 1346 out tokens · 29334 ms · 2026-05-18T03:44:42.634696+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

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

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

    quant-ph 2026-05 unverdicted novelty 5.0

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

Reference graph

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