The matrix Heinz mean and related divergence

Year 2022, Volume: 51 Issue: 2, 362 - 372, 01.04.2022

Trung Hoa Dınh Anh Vu Le Cong Trinh Le Ngoc Yen Phan

https://doi.org/10.15672/hujms.902879

Cited By: 2

Abstract

In this paper, we introduce a new quantum divergence
$$\Phi (X,Y) = \Tr \left[\left(\dfrac{1-\alpha}{\alpha}+ \dfrac{\alpha}{1-\alpha}\right)X+2Y - \dfrac{X^{1 -\alpha}Y^{\alpha}}{\alpha}- \dfrac{X^{\alpha}Y^{1-\alpha}}{1-\alpha} \right],$$
where $0< \alpha <1$.
We study the least square problem with respect to this divergence. We also show that the new quantum divergence satisfies the Data Processing Inequality in quantum information theory. In addition, we show that the matrix $p$-power mean $\mu_p(t, A, B) = ((1-t)A^p + tB^p)^{1/p}$ satisfies the in-betweenness property with respect to the new divergence.

Keywords

Quantum divergence, Heinz mean, least squares problems, matrix power mean, in-betweenness property, data processing inequality

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