No-Go Theorems for Quantum Transport Metrics from Fixed Cost Operators
arXiv:2607.28358
The paper proves that for quantum optimal transport based on fixed-cost operators, neither the optimal cost nor its square root can define a metric on the full quantum state space in dimensions three and higher, even for commuting states and under various stabilizations.
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 , 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 . Taken together, these no-go results show that fixed-cost coupling formulations do not lead to metrics on the full quantum state space.
17 pages