$α$-$z$-Rényi divergences in von Neumann algebras: data-processing inequality, reversibility, and monotonicity properties in $α,z$
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2024
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| author | Hiai, Fumio Jenčová, Anna |
| author_facet | Hiai, Fumio Jenčová, Anna |
| contents | We study the $α$-$z$-Rényi divergences $D_{α,z}(ψ\|φ)$ where $α,z>0$ ($α\ne1$) for normal positive functionals $ψ,φ$ on general von Neumann algebras, introduced in [S.~Kato and Y.~Ueda, arXiv:2307.01790] and [S.~Kato, arXiv:2311.01748]. We prove the variational expressions and the data processing inequality (DPI) for the $α$-$z$-Rényi divergences. We establish the sufficiency theorem for $D_{α,z}(ψ\|φ)$, saying that for $(α,z)$ inside the DPI bounds, the equality $D_{α,z}(ψ\circγ\|φ\circγ)=D_{α,z}(ψ\|φ)<\infty$ in the DPI under a quantum channel (or a normal $2$-positive unital map) $γ$ implies the reversibility of $γ$ with respect to $ψ,φ$. Moreover, we show the monotonicity properties of $D_{α,z}(ψ\|φ)$ in the parameters $α,z$ and their limits to the normalized relative entropy as $α\nearrow1$ and $α\searrow1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_07617 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $α$-$z$-Rényi divergences in von Neumann algebras: data-processing inequality, reversibility, and monotonicity properties in $α,z$ Hiai, Fumio Jenčová, Anna Quantum Physics Operator Algebras 81P45, 81P16, 46L10, 46L53, 94A17 We study the $α$-$z$-Rényi divergences $D_{α,z}(ψ\|φ)$ where $α,z>0$ ($α\ne1$) for normal positive functionals $ψ,φ$ on general von Neumann algebras, introduced in [S.~Kato and Y.~Ueda, arXiv:2307.01790] and [S.~Kato, arXiv:2311.01748]. We prove the variational expressions and the data processing inequality (DPI) for the $α$-$z$-Rényi divergences. We establish the sufficiency theorem for $D_{α,z}(ψ\|φ)$, saying that for $(α,z)$ inside the DPI bounds, the equality $D_{α,z}(ψ\circγ\|φ\circγ)=D_{α,z}(ψ\|φ)<\infty$ in the DPI under a quantum channel (or a normal $2$-positive unital map) $γ$ implies the reversibility of $γ$ with respect to $ψ,φ$. Moreover, we show the monotonicity properties of $D_{α,z}(ψ\|φ)$ in the parameters $α,z$ and their limits to the normalized relative entropy as $α\nearrow1$ and $α\searrow1$. |
| title | $α$-$z$-Rényi divergences in von Neumann algebras: data-processing inequality, reversibility, and monotonicity properties in $α,z$ |
| topic | Quantum Physics Operator Algebras 81P45, 81P16, 46L10, 46L53, 94A17 |
| url | https://arxiv.org/abs/2404.07617 |