The α-z-Bures Wasserstein divergence

In this paper, we introduce the α-z-Bures Wasserstein divergence for positive semidefinite matrices A and B as

where  is the matrix function in the α-z-Renyi relative entropy. We show that for 0≤ α ≤ z ≤ 1, the quantity Φ(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 μ(t, A, B)=((1 − t)A^p +tB^p)^1/p satisfies the in-betweenness property with respect to the α-z-Bures Wasserstein divergence.