Wigner symmetries single out symmetric Wasserstein distances in all finite dimensions

arXiv:2607.08298 · math-ph, math.MP, math.OA, quant-ph · Submitted 2026-07-09 · Read on arXiv

math-ph, math.MP, math.OA, quant-ph

Submitted: 2026-07-09

Updated: 2026-09-26

Terminology

• Wasserstein Distance — 64× in this paper · explained in 4 episodes
• Hilbert Space — 32× in this paper · explained in 2 episodes
• Optimal Transport — 16× in this paper · explained in 5 episodes
• Trace — 5× in this paper · explained in 2 episodes
• Gradient Flow — 2× in this paper · explained in 2 episodes
• Generalization — 1× in this paper · explained in 13 episodes
• Quantum Physics — 1× in this paper · explained in 11 episodes

Sources

• Wasserstein Distances on Quantum Structures: an Overview
• On the metric property of quantum Wasserstein divergences
• The swap transpose on couplings translates to Petz' recovery map on quantum channels
• Wasserstein distances and divergences of order $p$ by quantum channels
• Strong Kantorovich duality for quantum optimal transport with generic cost and optimal couplings on quantum bits
• Order $p$ quantum Wasserstein distances from couplings
• Quantum Wasserstein isometries of the $n$-qubit state space: a Wigner-type result
• Quantum Optimal Transport
• Quantum optimal transport is cheaper
• Towards Optimal Transport for Quantum Densities
• An Analog of the 2-Wasserstein Metric in Non-commutative Probability under which the Fermionic Fokker-Planck Equation is Gradient Flow for the Entropy
• Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance
• Non-commutative calculus, optimal transport and functional inequalities in dissipative quantum systems
• Relating relative entropy, optimal transport and Fisher information: a quantum HWI inequality
• Quantum optimal transport with quantum channels
• Quantum Optimal Transport: Quantum Channels and Qubits
• Optimal quantum channels
• Quadratic Wasserstein metrics for von Neumann algebras via transport plans
• Quantum Monge-Kantorovich problem and transport distance between density matrices
• On the Mean Field and Classical Limits of Quantum Mechanics

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