Monotonicity of a quantum 2-Wasserstein distance

• Published in: Journal of physics. A, Mathematical and theoretical Vol. 56; no. 9; pp. 95301 - 95324
• Main Authors: Bistroń, R, Eckstein, M, Życzkowski, K
• Format: Journal Article
• Language: English
• Published: IOP Publishing 03.03.2023
• Subjects: Monge–Kantorovich distance; monotonicity with respect to quantum channels; quantum transport problem
• ISSN: 1751-8113; 1751-8121
• DOI: 10.1088/1751-8121/acb9c8

Abstract We study a quantum analogue of the 2-Wasserstein distance as a measure of proximity on the set Ω N of density matrices of dimension N . We show that such (semi-)distances do not induce Riemannian metrics on the tangent bundle of Ω N and are typically not unitarily invariant. Nevertheless, we prove that for N  = 2 dimensional Hilbert space the quantum 2-Wasserstein distance (unique up to rescaling) is monotonous with respect to any single-qubit quantum operation and the solution of the quantum transport problem is essentially unique. Furthermore, for any N ⩾ 3 and the quantum cost matrix proportional to a projector we demonstrate the monotonicity under arbitrary mixed unitary channels. Finally, we provide numerical evidence which allows us to conjecture that the unitary invariant quantum 2-Wasserstein semi-distance is monotonous with respect to all CPTP maps for dimension N  = 3 and 4.