The α-z-Bures Wasserstein divergence

In this paper, we introduce the alpha-z-Bures Wasserstein divergence for positive semidefinite matrices A and B as Phi(A, B) = Tr((1-alpha)A + alpha B) - Tr(Q(alpha,z)(A, B)), where Q(alpha,z)(A, B) = (A(1- alpha/2z) B-alpha/z A(1-alpha/2z))(z) is the matrix function in the alpha-z-Renyi relative entropy. We show that for 0 <= alpha <= z <= 1, the quantity Phi(A, B) is a quantum divergence and satisfies the Data Processing Inequality in quantum information. We also solve the least squares problem with respect to the new divergence. In addition, we show that the matrix power mean mu(t, A, B) =((1 - t)A(p) + tB(p))(1/p) satisfies the in-betweenness property with respect to the alpha-z-Bures Wasserstein divergence. (C) 2021 Elsevier Inc. All rights reserved.

Automated summary

This paper introduces a new matrix divergence for positive semidefinite matrices, which satisfies the Data Processing Inequality in quantum information. The least squares problem with respect to the new divergence is also solved, and the property of the matrix power mean with respect to the divergence is explored.