The Quantum Wasserstein Distance of Order 1

TL;DR

This paper introduces a quantum version of the Wasserstein distance of order 1, providing a new metric for quantum states that generalizes classical distances and has applications in quantum information theory.

Contribution

It defines a quantum Wasserstein distance of order 1, proves a continuity bound for von Neumann entropy, and develops tools like quantum Lipschitz constants for quantum information analysis.

Findings

The quantum Wasserstein distance recovers classical distances in specific cases.

A continuity bound for von Neumann entropy is established.

Quantum transportation and concentration inequalities are proved.

Abstract

We propose a generalization of the Wasserstein distance of order 1 to the quantum states of $n$ qudits. The proposal recovers the Hamming distance for the vectors of the canonical basis, and more generally the classical Wasserstein distance for quantum states diagonal in the canonical basis. The proposed distance is invariant with respect to permutations of the qudits and unitary operations acting on one qudit and is additive with respect to the tensor product. Our main result is a continuity bound for the von Neumann entropy with respect to the proposed distance, which significantly strengthens the best continuity bound with respect to the trace distance. We also propose a generalization of the Lipschitz constant to quantum observables. The notion of quantum Lipschitz constant allows us to compute the proposed distance with a semidefinite program. We prove a quantum version of Marton's transportation inequality and a quantum Gaussian concentration inequality for the spectrum of quantum Lipschitz observables. Moreover, we derive bounds on the contraction coefficients of shallow quantum circuits and of the tensor product of one-qudit quantum channels with respect to the proposed distance. We discuss other possible applications in quantum machine learning, quantum Shannon theory, and quantum many-body systems.