Quantum Wasserstein Semimetric Overview

• Quantum Wasserstein semimetric is a noncommutative generalization of the classical Wasserstein metric that adapts to quantum states with features like nonzero self-distance.
• It is constructed via variational, geometric, and algebraic methods, employing formulations such as quantum couplings, projector-based cost operators, and fluid-dynamical minimization.
• The semimetric offers practical insights for quantum information, circuit complexity, and statistical learning while raising challenges in efficient computation and full metricification.

A quantum Wasserstein semimetric is a noncommutative generalization of the classical Wasserstein metric from optimal transport theory, adapted to the geometry of quantum states represented by density matrices or operator algebras. Unlike classical metrics—which are defined on probability measures and obey the triangle inequality—the quantum Wasserstein semimetric often only satisfies weaker forms of the metric axioms due to intrinsic quantum features such as non-commutativity, nonzero “self-distances,” and the possible failure of the triangle inequality in generality. It admits multiple rigorous constructions, several of which are inspired by variational, geometric, algebraic, or operational considerations, and has deep connections to quantum information theory, open quantum systems, statistical mechanics, noncommutative geometry, and quantum machine learning.

1. Fundamental Definitions and Constructions

The classical $p$-Wasserstein distance $W_p(\mu, \nu)$ between probability measures $\mu$ and $\nu$ is defined by

$$W_p^p(\mu, \nu) := \inf_{\pi \in \Gamma(\mu, \nu)} \int_{X \times X} d(x, y)^p \, d\pi(x, y),$$

where $\Gamma(\mu,\nu)$ is the set of couplings with marginals $\mu$ and $\nu$.

Quantum generalizations replace measures with quantum states (density matrices), and couplings with bipartite quantum states having fixed marginals or with quantum channels. The transport cost becomes an observable or operator acting on a tensor product space.

Canonical forms of the quantum Wasserstein semimetric include:

• Optimization over quantum couplings: For density matrices $\rho^A$, $\rho^B$ on Hilbert spaces $H_m$, $H_n$, and a Hermitian cost matrix $C$, the quantum optimal transport cost is

$$T_C(\rho^A, \rho^B) = \min_{\rho^{AB} \in \Gamma(\rho^A, \rho^B)} \mathrm{Tr}[C\, \rho^{AB}]$$

where $$\Gamma(\rho^A, \rho^B) = \{ \rho^{AB}: \mathrm{Tr}_B \rho^{AB} = \rho^A, \mathrm{Tr}_A \rho^{AB} = \rho^B \}$$ (Cole et al., 2021).

• Projector-based cost operators: A frequently used cost is the antisymmetric projector,

$C_Q = \frac{1}{2} (I - \mathrm{SWAP})$

yielding a semidistance $W(\rho^A,\rho^B) = \sqrt{T_{C_Q}(\rho^A, \rho^B)}$ (Cole et al., 2021, Bistroń et al., 2022).

• Fluid-dynamical/Brenier–Benamou-type approach: The quantum Wasserstein-2 semimetric is defined by minimizing the matricial “action” along curves $p(t)$ of density matrices:

$\int_0^1 \mathrm{tr}[p(t) v^*(t)v(t)]dt$

subject to a quantum continuity equation $ṗ = -\mathrm{div}_L(M_p(v))$ (Chen et al., 2016). Different matrix means for $M_p(v)$, e.g., anticommutator or logarithmic (Feynman–Kubo–Mori), change the induced geometry.

• Noncommutative $W_1$ norm on local perturbations: For $n$-qudit systems, one defines

$$W_1(\rho, \sigma) := \|\rho - \sigma\|_{W_1} = \frac{1}{2} \min \Big\{ \sum_{i=1}^n \|X^{(i)}\|_1 : \rho-\sigma = \sum_{i=1}^n X^{(i)},~\mathrm{Tr}_i X^{(i)} = 0 \Big\}$$

capturing the minimal “cost” in terms of single-qudit changes, generalizing the Hamming metric (Palma et al., 2020, Li et al., 2022).

• Gradient flow/geometric framework: Quantum Dirichlet forms and their first-order calculus define a noncommutative transport metric via the “action” on tangent directions weighted by operator means, recapturing classical $L^2$-Wasserstein geometry in commutative cases (Wirth, 2018, Becker et al., 2020).
• Husimi function/semiclassical setting: The quantum Wasserstein metric can be applied to Husimi functions. For density matrix $\rho$ and its Husimi function $_\rho(\alpha)$, one defines $W_p({}_{\mathcal{U}_T} \rho, \Phi_T {}_\rho)$, quantifying quantum-classical distance (Cotler et al., 8 Sep 2025).

2. Core Mathematical Properties and Metric Structure

Quantum Wasserstein semimetrics share certain key features:

• Non-negativity and symmetry: $W(\rho,\sigma)\ge 0$, $W(\rho,\sigma)=W(\sigma,\rho)$.
• Semimetric nature: Generically $W(\rho,\rho)\neq 0$ ("nonzero self-distance") for standard transport cost choices, though modified divergences can restore $W(\rho,\rho)=0$ via centering (Bunth et al., 20 Feb 2024, Bistroń et al., 2022). Triangle inequality may hold only under extra assumptions, or for specific cost models; proofs exist for certain classes of states (e.g., at least one pure) but require strong regularity in full generality (Bunth et al., 20 Feb 2024).
• Monotonicity under channels: For various constructions (notably the antisymmetric projector cost), monotonicity under CPTP maps,

$$W(\Phi(\rho^A), \Phi(\rho^B)) \leq W(\rho^A, \rho^B)$$

is satisfied (proven explicitly for qubits and mixed unitaries in higher dimension), making the semimetric suitable for assessing distinguishability under quantum processes (Bistroń et al., 2022).

• Riemannian structure and geodesics: For sufficiently regular choices (e.g., with logarithmic mean), the metric structure may admit geodesic paths and geodesic convexity of entropy, analogously to classical Wasserstein spaces, facilitating gradient flow and variational analyses (Wirth, 2018).
• Unitary/antiunitary isometries: For “symmetric” costs (involving all Pauli matrices), only Wigner-type symmetries—unitary or antiunitary conjugations—preserve the quantum Wasserstein divergence; for single-operator costs, the isometry group may be strictly larger, involving further norm-preserving affine transformations (Gehér et al., 2022, Simon et al., 19 Aug 2024).

Table: Key Properties of Various Quantum Wasserstein Semimetric Constructions

Approach / Reference	Self-distance $W(\rho,\rho)$	Triangle Inequality	Invariance
Antisymmetric projectors (Cole et al., 2021, Bistroń et al., 2022)	Generally $\ne 0$; can be centered	Holds for qubit; conjectured for $N>2$ projector costs	Unitary invariant for projectors
Noncommutative $W_1$ (Palma et al., 2020, Li et al., 2022)	$0$ iff $\rho=\sigma$	Yes	Permutations, 1-local unitaries
Dirichlet/gradient flow (Wirth, 2018)	$0$ iff $\rho = \sigma$	Satisfies on finite-entropy domain	Model dependent
Husimi/Wigner functions (Cotler et al., 8 Sep 2025)	Nonzero (semimetric)	See underlying classical $W_p$	Symplectic invariance

3. Quantum Benamou–Brenier and Fluid Dynamical Generalizations

Several constructions generalize the classical Benamou–Brenier dynamic (fluid) formulation:

• Continuity equation replacement: In the quantum setting, the probability density curve $p(t)$ becomes a curve of density matrices $p(t)$ or matrix-valued densities. The classical continuity equation $\partial_t p + \nabla \cdot (p v)=0$ is replaced by a quantum analogue:

$\dot{p} = -\mathrm{div}_L(M_p(v))$

$M_p(v)$ is a noncommutative “multiplication,” with key examples: - Anticommutator: $M_p(v) = pv + vp$ - Logarithmic mean: $M_p(v) = \int_0^1 p^{1-s} v p^s ds$ (the Feynman–Kubo–Mori mean) (Chen et al., 2016).

The corresponding action is

$$\min \int_0^1 \mathrm{tr}(p v^* v)\,dt \quad \text{subject to quantum continuity equation and boundary conditions}.$$

This formalism captures quantum analogues of kinetic energy, “quantum velocities,” and continuity, and leads to convex minimization problems (for instance, upon introducing a quantum momentum variable $u=p v$) (Chen et al., 2016).

When logarithmic mean is used, the entropy gradient flow becomes linear (quantum heat equation), mirroring the classical case; for anticommutator, the flow is nonlinear and inherently quantum (Chen et al., 2016).

4. Geometric and Information-Theoretic Implications

The quantum Wasserstein semimetric framework leads to new geometric structures and insights:

• Quantum gradient flows: The gradient flow of quantum entropy $S(p) = -\mathrm{tr}(p\log p)$ with respect to matricial Wasserstein geometries yields dissipative or heat-type equations (e.g., $ṗ = \Delta_L p$ for the logarithmic case) (Chen et al., 2016, Wirth, 2018).
• Synthetic Ricci curvature: In the noncommutative gradient-flow framework, the convexity of entropy along geodesics induced by the quantum Wasserstein metric encodes lower Ricci curvature bounds analogously to Lott–Villani–Sturm theory (Wirth, 2018).
• Talagrand-type inequalities: Under convexity/geometric assumptions,

$W(\rho, 1)^2 \leq \frac{2}{K} \mathrm{Ent}(\rho),$

generalizes classical transport-entropy inequalities (Wirth, 2018).

• Relation to Wigner–Yanase metric: Self-transport costs encode quantum fluctuations, linking the quantum Wasserstein semimetric to quantum information geometry (Palma et al., 2019).
• Lipschitz observables and duality: The dual formulations (quantum Kantorovich–Rubinstein duality) yield quantum Lipschitz constants for observables, central for the analysis of concentration inequalities and for operational interpretations (e.g., estimation and learning tasks) (Palma et al., 2020, Duvenhage et al., 2022).

5. Operational and Physical Applications

Quantum Wasserstein semimetrics have demonstrated wide applicability:

• Quantum information and quantum computation: Robust loss functions in quantum generative adversarial networks (qWGANs), with quantum Wasserstein metrics enabling stable adversarial training and circuit compression (Chakrabarti et al., 2019).
• Quantum circuit complexity: Wasserstein complexity, defined as

$C_{W_1}(U) = \max_{\rho} W_1(\rho, U\rho U^*)$

provides lower bounds on both theoretical and experimental resource requirements of implementing $U$ as a quantum circuit (Li et al., 2022). Faithfulness, convexity, and subadditivity properties ensure the operational relevance of the metric.

• Mean-field and semiclassical analysis: Optimal transport inequalities using quantum Wasserstein distances provide precise rates of convergence from quantum to classical dynamics (e.g., Egorov-type theorems in Husimi function representation), essential for semiclassical and many-body problems (Cotler et al., 8 Sep 2025).
• Quantum Markov semigroups & open system dynamics: The geometry induced by the quantum Wasserstein semimetric enables analysis of entropy production, mixing, and convergence rates for quantum dynamics (Chen et al., 2016, Wirth, 2018).
• Noncommutative geometry and quantum groups: Quantum Wasserstein distances constructed over noncommutative or “quantum” permutation groups extend the classical theory of metric spaces to operator algebras, supporting the paper of quantum symmetries and noncommutative geometry (Anshu et al., 25 May 2025).
• Quantum statistical learning: Natural gradient flows using the quantum Wasserstein information matrix provide Riemannian descent schemes for statistical estimation and state tomography, both in finite- and infinite-dimensional settings (Becker et al., 2020).

6. Extensions, Variants, and Contemporary Results

Recent literature has extended the notion of quantum Wasserstein semimetrics in several directions:

• Wasserstein divergences and "faithful" metrics: By centering the cost to remove nonzero self-distances (i.e., $$d_A(\rho,\omega) = \sqrt{ D_A^2(\rho,\omega) - \frac12( D_A^2(\rho,\rho)+D_A^2(\omega,\omega))}$$), a bona fide metric (including the triangle inequality) can be rigorously established under finite-energy and purity conditions, and strong numerical evidence supports validity for general mixed states (Bunth et al., 20 Feb 2024).
• Quantum Wasserstein distances of arbitrary order $p$: Using coupling-based formulations, the distances

$$W_p^d(\rho,\sigma) = \left( \inf_{Q \in \mathcal{Q}(\rho,\sigma)} \sum_j q_j d(|\psi_j\rangle,|\phi_j\rangle)^p \right )^{1/p}$$

unify and extend to general metrics $d$ on pure states, interpolating between trace distance, Hamming-type, and complexity metrics. This offers a flexible operational toolkit for both theoretical and applied quantum information (Beatty et al., 26 Feb 2024).

• Isometry groups and symmetries: Exact characterizations of isometries for various models (e.g., the symmetric cost vs. single Pauli observable) reveal that for full Pauli costs, Wigner-type symmetries (unitary or antiunitary conjugations) exhaust the isometry group, while for reduced costs, further affine or flip symmetries may appear (Gehér et al., 2022, Simon et al., 19 Aug 2024).
• Noncommutative channel metrics: Quantum Wasserstein metrics have been extended to quantum channels, constructed via additivity and reductions on subsystems, with explicit operator algebraic gauge norms capturing locality and stability under composition and tensorization (Duvenhage et al., 2022).

7. Open Problems and Future Directions

The theory of quantum Wasserstein semimetrics continues to develop, with key ongoing directions including:

• Full metricification: Determining the most general conditions under which quantum Wasserstein divergences become metrics, both analytically and numerically (Bunth et al., 20 Feb 2024).
• Efficient computation: Algorithms for computing quantum Wasserstein distances of higher order or for high-dimensional/many-body systems remain a challenge, especially given the necessity of solving semidefinite programs or optimizing over joint quantum states.
• Extensions to infinite dimensions and field theory: Adapting the framework to quantum field theory and infinite-dimensional systems (e.g., lattices, continuous variables) raises new technical and conceptual questions (Palma et al., 2022).
• Quantum optimal transport in noncommutative geometry: Further development of quantum metrics for operator algebras and quantum groups, examining spectral triples and metric convergence in the noncommutative Gromov–Hausdorff sense (Anshu et al., 25 May 2025).
• Physical resource measures: Exploiting quantum Wasserstein semimetrics for quantifying circuit complexity, state discrimination, privacy, and characterizing resource theories in quantum computation and thermodynamics (Li et al., 2022).

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Quantum Wasserstein semimetrics synthesize optimal transport, noncommutative geometry, information theory, and dissipative quantum dynamics, providing both new mathematical structures for state and channel comparison and a wealth of applications across quantum science. Their variational formulations, geometric underpinnings, and operational interpretations underpin modern developments in quantum statistical mechanics, learning, complexity, and noncommutative analysis, while raising rigorous foundational and computational challenges that drive current research.