Abstract
In this paper, we study a matrix equation involving the matrix geometric mean \(A\natural_{t}B=A^{1/2}(A^{-1/2}BA^{-1/2})^{t}A^{1/2},\) \(t\in(1,2)\). We study several properties and inequalities for the unique solution of such an equation. We also use the weighted geometric mean \(A\natural_{t}B\) to define a new quantum Hellinger divergence and show that the new quantum divergence satisfies the Data Processing Inequality.
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ACKNOWLEDGMENTS
We would like to express our sincere thanks to Professor Grigori Amosov for careful reading and multiple valuable comments that helped improve the current manuscript.
Funding
For the second author, this research is funded by Vietnam National University Ho Chi Minh City (VNU-HCM) under grant number B2023-34-01. The first author is funded by a Troy University research grant.
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Dinh, T.H., Le, A.V., Nguyen, T. et al. A Generalized Matrix Power Mean and a New Quantum Hellinger Divergence. Lobachevskii J Math 45, 636–647 (2024). https://doi.org/10.1134/S1995080224600304
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DOI: https://doi.org/10.1134/S1995080224600304