The Quantum Wasserstein Semidistance Is Not a Distance
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Statement
Friedland and coauthors proposed a quantum analogue of the -Wasserstein distance and conjectured that, though only a semidistance in general, it is a true distance for a particular quantum cost matrix and for cost matrices near it. The paper disproves both conjectures with an explicit family of triples of states violating the triangle inequality.
Context
Refutes a conjecture from a Physical Review Letters paper about the metric structure of quantum optimal transport.
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The acknowledgements state the counterexamples were found with the assistance of ChatGPT 5.
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