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.
📌 3 Key Takeaways
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Objective & Hypothesis: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.
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Key Findings & Data:We also show that the new quantum divergence satisfies the Data Processing Inequality in quantum information theory.
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Conclusion & Impact: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.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Quantum divergence · Heinz mean · least squares problems
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.
DINH, T. H., LE, A. V., LE, C. T., PHAN, N. Y. (2022). The matrix Heinz mean and related divergence. Hacettepe Journal of Mathematics and Statistics, Cilt: 51 - Sayı: 2, 362–372. https://ajindex.com/articles/669212
J. Pitrik and D. Virosztek, A divergence center interpretation of general symmetric Kubo-Ando means, and related weighted multivariate operator means, Linear Algebra Appl. 609, 203-217, 2021.