Quantum Wasserstein Distances

• Quantum Wasserstein distances are noncommutative generalizations of optimal transport metrics, quantifying cost for transporting quantum states via channel-based formulations.
• They employ coupling-based definitions and semidefinite programming techniques to ensure metric properties and compute distances in finite-dimensional quantum systems.
• Applications span quantum machine learning, many-body physics, and quantum communication, providing operational and geometric insights into quantum information theory.

Quantum Wasserstein distances generalize the classical Wasserstein metric, central to optimal transport theory, to the noncommutative setting of quantum states, channels, and operator algebras. In classical settings, the Wasserstein distance quantifies the minimal cost to transport one probability measure to another with respect to a prescribed cost function. Quantum generalizations inherit and extend these principles by leveraging the structure of quantum channels, convex optimization over bipartite couplings, the duality between observable Lipschitz constraints and state transport, and provide geometric and operational perspectives relevant for quantum information theory, quantum machine learning, and functional analysis.

1. Formulation of Quantum Wasserstein Distances

In quantum optimal transport, the p-Wasserstein distance between quantum states $\rho$, $\omega$ on Hilbert space $\mathcal{H}$ is constructed by minimizing a transport cost over admissible quantum couplings or quantum channels:

• Coupling-based definition: For observables $A = \{A_1, \ldots, A_K\}$ and $p \geq 1$, the distance is defined by optimizing over quantum couplings $\Pi \in S(\mathcal{H} \otimes \mathcal{H}^*)$ with partial traces $\operatorname{tr}_{\mathcal{H}^*} \Pi = \omega$ and $\operatorname{tr}_{\mathcal{H}} \Pi = \rho^{T}$,

$$D^{(A)}_{p,p}(\rho, \omega) = \left( \inf_{\Pi \in C(\rho, \omega)} \operatorname{tr}\left[\Pi^T C^{(A)}_{p,p} \right] \right)^{\min \{1, 1/p\}}$$

where the cost operator is $$C^{(A)}_{p,p} = \sum_k (A_k \otimes I^{T} - I \otimes A_k^T)^p$$.

• Channel-based realization: Quantum channels (completely positive trace-preserving maps) act as transport plans, providing a physically operational interpretation consistent with information-theoretic tasks (Palma et al., 2019).
• Quadratic divergences: For $p=2$, the quadratic quantum Wasserstein divergence is modified to ensure metric properties. If $D_A^2(\rho, \omega)$ is the squared cost with respect to observables $A$,

$$d_A(\rho, \omega) = \sqrt{ D_A^2(\rho, \omega) - \frac{1}{2} [ D_A^2(\rho, \rho) + D_A^2(\omega, \omega) ] }$$

This correction eliminates nonzero “self-distances” characteristic of unmodified quantum Wasserstein definitions (Bunth et al., 20 Feb 2024, Bunth et al., 14 Jan 2025).

2. Geometric and Metric Properties

Quantum Wasserstein distances exhibit a rich geometric structure, reflecting both quantum information-theoretic and optimal transport features:

• Modified triangle inequality: Unlike classical Wasserstein distances, certain quantum distances satisfy only a modified triangle inequality, typically involving an extra “self-distance” term, e.g.

$$D(\rho_A, \rho_C) \leq D(\rho_A, \rho_B) + D(\rho_B, \rho_B) + D(\rho_B, \rho_C)$$

This property arises due to quantum fluctuation-induced nonzero self-costs (Palma et al., 2019).

• True metric behavior: Quadratic divergences and various separable-coupling variants restore genuine metric properties (distance zero only for identical states, symmetric, triangle inequality holds) at least for qubits and for arbitrary states when one state is pure (Bunth et al., 20 Feb 2024, Tóth et al., 17 Jun 2025).
• Relation to quantum Fisher information: For appropriately chosen cost operators and optimization over separable states, the self-distance of a quantum state is proportional to its quantum Fisher information: $D(\rho, \rho)^2 = \frac{1}{4} F_Q[\rho, H]$ (Tóth et al., 2022, Tóth et al., 17 Jun 2025). This connects quantum transport to quantum metrology.
• Additivity, stability properties: For order-1 distances extended to channels, additivity under tensor products and stability under trivial subsystem operations are established in a general operator algebraic framework (Duvenhage et al., 2022).

3. Variants and Extensions

Multiple formulations exist, each tailored to preserve different aspects of classical optimal transport or to accommodate quantum-specific constraints:

Approach	Admissible Couplings	Cost Operator Examples
Coupling with all bipartite states	All density matrices on $\mathcal{H} \otimes \mathcal{H}^*$	$(A_k \otimes I^{T} - I \otimes A_k^T)^p$, swap-based, projectors
Coupling with separable states	Only separable (non-entangled) bipartite states	Same as above
Quantum channels between states	CPTP maps with prescribed marginals	Induced cost via Choi-Jamiolkowski correspondence

Certain definitions recover classical Wasserstein distances for diagonal (commutative) states, are invariant under symmetric operations, and are additive with respect to tensor products (Palma et al., 2020, Duvenhage et al., 2022, Anshu et al., 25 May 2025).

Dynamical approaches mimic the Benamou–Brenier perspective by defining a Riemannian structure on quantum state manifolds via quantum Markov semigroups (Beatty, 11 Jun 2025).

4. Computational Methods

Computing quantum Wasserstein distances is generally nontrivial due to infinite-dimensional optimization and noncommutativity:

• Moment-SOS hierarchy: For order-2 distances, the problem can be encoded as an infinite-dimensional linear program over positive measures on products of unit spheres, handled computationally by the moment-sums of squares hierarchy—yielding converging sequences of certified lower bounds via semidefinite programming (Chhatoi et al., 24 Jun 2025).
• Semidefinite programming for order-1 distances: The dual characterization via quantum Lipschitz constants enables efficient computation of the quantum $W_1$ distance using SDP techniques, especially for finite-dimensional systems (Palma et al., 2020, Duvenhage et al., 2022, Palma et al., 2022).
• Quantum annealing approaches: QUBO formulations on D-Wave quantum annealers have been used for matching-type Wasserstein computations in topological data analysis contexts (Berwald et al., 2018).

5. Applications in Quantum Information and Physics

Quantum Wasserstein distances have found applications in multiple domains:

• Quantum machine learning: Quantum Wasserstein-type loss functions mitigate challenges like barren plateaus by encoding local changes, improving training stability in quantum GANs and state estimation tasks (Kiani et al., 2021).
• Many-body physics: Wasserstein metrics display critical scaling near quantum phase transitions (e.g., in the transverse field Ising model), directly connecting quantum criticality and correlation functions to the transport cost (Camacho et al., 3 Apr 2025).
• Quantum permutation groups and symmetry analysis: Quantum analogues of the Hamming/Wasserstein distance on permutation groups generalize classical metric and convolution properties, illuminating symmetries and Lipschitz element density in associated C*-algebras (Anshu et al., 25 May 2025).
• Quantum statistical mechanics: Bounds on energy fluctuations, entropy continuity, and transportation-cost inequalities are established in terms of Wasserstein distances, yielding uniqueness of Gibbs states under certain interaction regimes (Palma et al., 2022).
• Quantum communication and complexity metrics: Variants allow natural operational extensions (e.g., Nielsen's complexity metric) from pure to mixed states (Beatty et al., 26 Feb 2024).

6. Controversies and Open Questions

The quantum Wasserstein literature reveals significant diversity in definitions and properties:

• There is no universally agreed-upon “true” quantum Wasserstein distance; various frameworks prioritize different geometric, operational, or metrical aspects (Beatty, 11 Jun 2025).
• Some definitions do not satisfy the triangle inequality or yield nonzero self-distances, necessitating corrective divergences or particular restrictions on the class of couplings.
• The precise quantitative relationships and bounds among coupling-, dynamical-, and Lipschitz-based formulations remain under active investigation.
• Efficient numerical approximations, extensions to infinite-dimensional and open quantum systems, and understanding the sensitivity to various choices of cost operators or coupling constraints are open areas of research.

7. Future Directions

Research continues in linking classical and quantum optimal transport, extending computationally tractable methods to larger quantum systems, exploiting transport metrics in quantum learning algorithms (notably for phase recognition and state discrimination), and leveraging the transport-geometric perspective for quantum resource theories and statistical mechanics.

Current lines of inquiry include:

• Establishing equivalence and bounds between different quantum Wasserstein frameworks.
• Bridging quantum Wasserstein distances with physical operational tasks, such as channel discrimination and adversarial learning.
• Characterizing the symmetries and isometry groups associated with various quantum Wasserstein metrics (Simon et al., 19 Aug 2024).
• Developing scalable algorithms for Wasserstein computations in noncommutative and high-dimensional quantum systems.

The field continues to evolve at the intersection of quantum information, mathematical physics, and optimal transport theory, with quantum Wasserstein distances emerging as a unifying tool for quantifying state and channel similarity, robustness, and complexity in noncommutative spaces.