A quantum version of the Monge-Kantorovich optimal transport problem is analyzed. The transport cost is minimized over the set of all bipartite coupling states ρ^{AB} such that both of its reduced density matrices ρ^{A} and ρ^{B} of dimension N are fixed. We show that, selecting the quantum cost matrix to be proportional to the projector on the antisymmetric subspace, the minimal transport cost leads to a semidistance between ρ^{A} and ρ^{B}, which is bounded from below by the rescaled Bures distance and from above by the root infidelity. In the single-qubit case, we provide a semianalytic expression for the optimal transport cost between any two states and prove that its square root satisfies the triangle inequality and yields an analog of the Wasserstein distance of the order of 2 on the set of density matrices. We introduce an associated measure of proximity of quantum states, called swap fidelity, and discuss its properties and applications in quantum machine learning.
Coupling-based quantum optimal transport generalizes classical optimal transport by representing transport plans as bipartite states with prescribed marginals and evaluating their cost as the expectation of a fixed Hermitian operator. Friedland et al. [Phys. Rev. Lett. 129, 110402 (2022)] conjectured that the square root of the optimal cost associated with the SWAP projector is a metric in every dimension and that this property persists for nearby quantum cost matrices. Miller [arXiv:2607.07764] disproved both conjectures by constructing explicit diagonal qutrit counterexamples. Building on his analysis, we prove a uniform no-go theorem for standard couplings. In every dimension
d≥3, no fixed cost operator makes either the optimal cost or its square root a metric, with violations occurring already among commuting states. The obstruction persists under stabilization of the SWAP cost. For channel-induced couplings, global nonnegativity and vanishing self-cost force the cost operator to be zero, precluding point separation when
d≥2. Taken together, these no-go results show that fixed-cost coupling formulations do not lead to metrics on the full quantum state space.
Quantum Wasserstein divergences are modified versions of quantum Wasserstein distances defined by channels and they have been conjectured to be genuine metrics on quantum state spaces by De Palma and Trevisan. We prove triangle inequality for quantum Wasserstein divergences for every quantum system described by a separable Hilbert space and any quadratic cost operator under the assumption that a particular state involved is pure and all the states have finite energy. We also provide strong numerical evidence suggesting that the triangle inequality holds in general for an arbitrary choice of states.
Convex roofs are natural ways to extend a scalar function defined on the extreme points of a convex set
K to the whole set. In this work we focus on the case
K=C×C for some convex set
C, and the extension of a distance
d defined on the extreme points of
C. In general, the triangle inequality may not survive the extension process. We therefore consider the metric envelope of such extensions and study the metric properties they endow
C with. More generally, we also consider products of different convex sets
C1×C2 and costs defined on their extreme points, in which case no metric envelope is involved. We then show a two-way correspondence between convex extensions and Kantorovich optimal transport formulations. On the one hand, these extensions can be recovered as quotients of standard, classical Kantorovich costs or Wasserstein distances by the barycenter map. On the other hand, various Kantorovich formulations re-write as such convex extensions. Classical optimal transport is recovered as the case of the simplex, and both the semiclassical optimal transport cost of Golse and Paul and the entanglement-free quantum Wasserstein objects of Beatty and Stilck França are recovered by considering the non-simplex convex set of quantum states.
This work provides a clear bridge between the theory of convex extensions and that of optimal transport, with a unifying framework for classical, semiclassical and entanglement-free quantum instances of Kantorovich formulations.
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.
We develop a Quantum Optimal Transport (QOT) barycenter framework for quantum states, as an analog of Wasserstein barycenters~\cite{AguCar}, and establish existence and duality results for a broad class of possibly unbounded transport costs on separable Hilbert spaces. Our framework provides, in particular, a unified treatment of
2-quantum Wasserstein (QW) barycenters in both the quantum-state and quantum-channel formulations by specializing to the canonical quadratic cost operators associated with the
2-quantum Wasserstein distances of Caglioti--Golse--Mouhot--Paul~\cite{caglioti2021towards,Golse-Mouhot-Paul:2016} and De Palma--Trevisan~\cite{DPaTre19}.
The central results concern Gaussian input states and, in particular, Gaussian rigidity: whether Gaussian input states force the 2-QW barycenter itself to be Gaussian and uniquely determined when the minimization is taken over all quantum states. We first show that the barycenter problem admits a Gaussian minimizer and reduces to a finite-dimensional convex optimization problem over covariance matrices.
The main difficulty is that uniqueness of the optimal covariance does not, in general, imply uniqueness of the underlying quantum state. We bridge this gap through a state-reconstruction principle under covariance complementary slackness that upgrades uniqueness of the optimal covariance to uniqueness of the full quantum state, and thereby prove a global rigidity theorem: if at least one Gaussian input is faithful, then the barycenter is unique among all quantum states and is necessarily Gaussian. Faithfulness is sufficient but not necessary: families of pure inputs still determine a unique barycenter, whereas partially pure nonfaithful Gaussian inputs may admit multiple barycenters.