No-Go Theorems for Quantum Transport Metrics from Fixed Cost Operators

• URL: http://arxiv.org/abs/2607.28358v1
• Date: Thu, 30 Jul 2026 15:24:50 GMT
• Title: No-Go Theorems for Quantum Transport Metrics from Fixed Cost Operators
• Abstract summary: Coupling-based quantum optimal transport generalizes classical optimal transport.<n>No-go results show that fixed-cost coupling formulations do not lead to metrics on the full quantum state space.
• Score: 2.574393697080417
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 $d\geq3$, 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 $d\geq2$. Taken together, these no-go results show that fixed-cost coupling formulations do not lead to metrics on the full quantum state space.

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