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arxiv: 2506.14523 · v6 · pith:E3JZVFO6new · submitted 2025-06-17 · 🪐 quant-ph · math-ph· math.MP

Quantum Wasserstein distance and its relation to several types of fidelities

G\'eza T\'oth , J\'ozsef Pitrik This is my paper

Pith reviewed 2026-05-19 09:06 UTC · model grok-4.3

classification 🪐 quant-ph math-phmath.MP
keywords quantum Wasserstein distanceUhlmann-Jozsa fidelityseparable statessuperfidelitytriangle inequalitybipartite quantum statesquantum distances
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The pith

Several definitions of the quantum Wasserstein distance become identical when the optimization is restricted to separable states, and equal the square root of the Uhlmann-Jozsa fidelity for qubits.

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

The paper examines multiple ways to define a quantum Wasserstein distance by optimizing over bipartite quantum states that share the same marginals on each side. It then shows what happens when that optimization is limited to separable states rather than arbitrary states. Several of the resulting quantities turn out to be equal to one another, linking previously separate approaches in the literature. The same restriction also lets the square root of the standard Uhlmann-Jozsa fidelity be expressed as an optimization over separable states. For qubits this produces exact equality with the fidelity, and further relations appear with the superfidelity.

Core claim

When the optimization that defines the quantum Wasserstein distance is carried out only over separable bipartite states with fixed marginals, several distinct quantities coincide. This common value also equals the square root of the Uhlmann-Jozsa quantum fidelity when the states are qubits. The same separable-state formulation yields the triangle inequality for some of the quantities whenever one of the three states under comparison is pure, and it produces explicit relations with the superfidelity.

What carries the argument

The restriction of the defining optimization for quantum Wasserstein distance to separable bipartite states with given marginals, which produces equalities among several candidate distances and an optimization expression for the square root of Uhlmann-Jozsa fidelity.

If this is right

  • Multiple previously distinct quantum distance measures become identical under the separable restriction.
  • The square root of Uhlmann-Jozsa fidelity admits an explicit representation as a separable-state optimization.
  • Equality between the distance and the fidelity holds exactly on qubit systems.
  • A triangle inequality is obtained for some of the quantities when one state is pure.
  • Additional relations connect the quantities to the superfidelity.

Where Pith is reading between the lines

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

  • The separable-state formulation may simplify numerical evaluation of these distances in low-dimensional systems.
  • The same restriction could be tested on higher-dimensional states to see whether the equalities persist beyond qubits.
  • The approach suggests a route to derive further inequalities by combining the separable optimization with known properties of fidelity.

Load-bearing premise

The optimization over general bipartite states with given marginals remains meaningful and produces the same relations when it is restricted to separable states.

What would settle it

An explicit pair of qubit states for which the value obtained by optimizing over separable states differs from the square root of their Uhlmann-Jozsa fidelity.

read the original abstract

We consider several definitions of the quantum Wasserstein distance based on an optimization over general bipartite quantum states with given marginals. Then, we examine the quantities obtained after the optimization is carried out over bipartite separable states instead. We prove that several of these quantities are equal to each other. Thus, we connect several approaches in the literature. We prove the triangle inequality for some of these quantities for the case of one of the three states being pure. As a byproduct, we show that the square root of the Uhlmann-Jozsa quantum fidelity can also be written as an optimization over separable states with given marginals. We use this to prove that some of these quantities equal the Uhlmann-Jozsa quantum fidelity for qubits. We also find relations with the superfidelity.

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 considers several definitions of quantum Wasserstein distance obtained via optimization over general bipartite quantum states with fixed marginals. It then restricts the optimization to separable states, proves that multiple resulting quantities coincide with one another, establishes a triangle inequality when one of the three states is pure, shows that the square root of the Uhlmann-Jozsa fidelity admits an equivalent formulation as an optimization over separable states with given marginals, and demonstrates that certain of these quantities equal the Uhlmann-Jozsa fidelity for qubits. Relations to the superfidelity are also derived.

Significance. If the proofs are correct, the work unifies several existing approaches to quantum Wasserstein distances by exhibiting their equivalence under the separable restriction. The separable-state representation of sqrt(Uhlmann-Jozsa fidelity) supplies a new variational characterization that may prove useful for computation or interpretation. The triangle inequality for the pure-state case and the qubit equalities are obtained via direct algebraic identities and case-by-case verification, which strengthens verifiability. These results connect disparate strands of the literature on quantum optimal transport and fidelity measures.

major comments (1)
  1. [Triangle inequality section] The triangle inequality is established only under the assumption that one of the three states is pure. Clarify whether this restriction is essential for the proof technique or whether a general mixed-state version can be obtained within the same framework; this directly affects how broadly the quantities can be regarded as distances.
minor comments (2)
  1. [Definitions and main results] Define the various Wasserstein-type quantities with explicit labels (e.g., W_sep, W_gen) at their first appearance and maintain consistent notation when stating the equalities.
  2. [Qubit case] In the qubit equality proofs, include the explicit matrix representations or parameterizations used for the states to facilitate direct verification.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and valuable feedback on our manuscript. We address the major comment in detail below.

read point-by-point responses

  1. Referee: [Triangle inequality section] The triangle inequality is established only under the assumption that one of the three states is pure. Clarify whether this restriction is essential for the proof technique or whether a general mixed-state version can be obtained within the same framework; this directly affects how broadly the quantities can be regarded as distances.

    Authors: The restriction to the case where one of the three states is pure is essential to our proof technique. Our proof makes use of specific properties that hold when one state is pure, such as the ability to express the state in a form that simplifies the optimization over separable states and allows direct verification of the triangle inequality through algebraic manipulation. Extending this to general mixed states does not appear straightforward within the current framework, and we have not identified a general mixed-state triangle inequality. We will add a clarifying remark in the revised version of the manuscript to explicitly state this limitation of the proof technique and note that the general case remains open. This addresses the referee's concern without altering the main results. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations rest on explicit proofs and algebraic identities

full rationale

The paper defines quantum Wasserstein distances via optimization over bipartite states with fixed marginals, then restricts the optimization to separable states and proves equalities between several resulting quantities using direct algebraic manipulation and case-by-case verification. It further shows that the square root of Uhlmann-Jozsa fidelity admits an equivalent formulation as a separable-state optimization and establishes equality with fidelity for qubits. The triangle inequality is proved directly for the pure-state case. No step reduces a claimed prediction or central result to a fitted parameter, a self-referential definition, or a load-bearing self-citation whose validity depends on the present work. All equalities follow from the stated optimization problems and standard properties of quantum states without importing unverified uniqueness theorems or ansatzes from the authors' prior publications. The derivation chain is therefore self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on standard definitions and properties from quantum information theory without introducing new free parameters or invented entities; the central claims rest on mathematical properties of bipartite states and optimizations.

axioms (2)
  • standard math Bipartite quantum states possess well-defined marginals and the set of separable states is a proper subset of all states.
    Invoked when restricting the optimization from general to separable states.
  • domain assumption The Uhlmann-Jozsa fidelity and superfidelity are standard measures with known properties in quantum mechanics.
    Used when relating the optimized quantities to fidelity.

pith-pipeline@v0.9.0 · 5665 in / 1365 out tokens · 73474 ms · 2026-05-19T09:06:51.194294+00:00 · methodology

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Forward citations

Cited by 2 Pith papers

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

  1. General method for obtaining the energy minimum of spin Hamiltonians for separable states

    quant-ph 2026-05 unverdicted novelty 6.0

    A method is given to compute the minimum energy of certain spin Hamiltonians over separable states, expressed via quantum Fisher information for Ising models and fidelity for Heisenberg chains.

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

    math-ph 2025-10 unverdicted novelty 6.0

    Establishes Kantorovich duality for linearized non-quadratic quantum optimal transport realized by channels, determines optimal primal-dual solutions for qubits under state restrictions, and proves the triangle inequa...

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