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arXiv · 2103.09893

Monotonic multi-state quantum $f$-divergences

Keiichiro Furuya·Nima Lashkari·Shoy Ouseph
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Abstract

We use the Tomita-Takesaki modular theory and the Kubo-Ando operator mean to write down a large class of multi-state quantum $f$-divergences and prove that they satisfy the data processing inequality. For two states, this class includes the $(\alpha,z)$-R\'enyi divergences, the $f$-divergences of Petz, and the measures in \cite{matsumoto2015new} as special cases. The method used is the interpolation theory of non-commutative $L^p_\omega$ spaces and the result applies to general von Neumann algebras including the local algebra of quantum field theory. We conjecture that these multi-state R\'enyi divergences have operational interpretations in terms of the optimal error probabilities in asymmetric multi-state quantum state discrimination.

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Keiichiro Furuya, Nima Lashkari, Shoy Ouseph. 2021-03-17. Monotonic multi-state quantum $f$-divergences. https://doi.org/10.1063/5.0125505

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