Relations between different definitions of the quantum Wasserstein distance for qubits

TL;DR

This paper shows that for qubits, different quantum Wasserstein distance definitions align, and the self-distance relates to Wigner-Yanase skew information, clarifying their connection.

Contribution

It demonstrates the equivalence of two quantum Wasserstein distances for qubits and links the self-distance to Wigner-Yanase skew information.

Findings

Quantum Wasserstein distances coincide for qubits with a single operator in the cost function.

Self-distance equals Wigner-Yanase skew information for qubits.

Clarifies the relationship between different quantum distance measures.

Abstract

The quantum Wasserstein distances defined by Golse, Mouhot, Paul, and Caglioti and by De Palma and Trevisan coincide for qubits when a single operator appears in the cost function. As a consequence, the self-distance equals the Wigner-Yanase skew information in this case.