The matrix Heinz mean and related divergence

Authors : Trung Hoa DINH, Anh Vu LE, Cong Trinh LE, Ngoc Yen PHAN

Pages : 362-372

Doi:10.15672/hujms.902879

View : 14 | Download : 7

Publication Date : 2022-04-01

Article Type : Research Paper

Abstract :In this paper, we introduce a new quantum divergence $$\Phi insert ignore into journalissuearticles values(X,Y); = \Tr \left[\leftinsert ignore into journalissuearticles values(\dfrac{1-\alpha}{\alpha}+ \dfrac{\alpha}{1-\alpha}\right);X+2Y - \dfrac{X^{1 -\alpha}Y^{\alpha}}{\alpha}- \dfrac{X^{\alpha}Y^{1-\alpha}}{1-\alpha} \right],$$ 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 $\mu_pinsert ignore into journalissuearticles values(t, A, B); = insert ignore into journalissuearticles values(insert ignore into journalissuearticles values(1-t);A^p + tB^p);^{1/p}$ 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