Quantum Wasserstein distance and its relation to several types of fidelities

TL;DR

This paper explores various quantum Wasserstein distances, establishing their equivalences, and relates them to quantum fidelities and superfidelity, with specific results for qubits.

Contribution

It introduces and compares different quantum Wasserstein distances, proving their equivalences and connections to quantum fidelities, including for separable states and qubits.

Findings

Several quantum Wasserstein distances are shown to be equal.

Triangle inequality proven for cases with one pure state.

Square root of Uhlmann-Jozsa fidelity expressed as an optimization over separable states.

Abstract

We consider several definitions of the quantum Wasserstein distance based on an optimization over general bipartite quantum states with given marginals. Then, we examine the quantities obtained after the optimization is carried out over bipartite separable states instead. We prove that several of these quantities are equal to each other. Thus, we connect several approaches in the literature. We prove the triangle inequality for some of these quantities for the case of one of the three states being pure. As a byproduct, we show that the square root of the Uhlmann-Jozsa quantum fidelity can also be written as an optimization over separable states with given marginals. We use this to prove that some of these quantities equal the Uhlmann-Jozsa quantum fidelity for qubits. We also find relations with the superfidelity.