\(\alpha \)-z-Rényi Divergences in von Neumann Algebras: Data Processing Inequality, Reversibility, and Monotonicity Properties in \(\alpha ,z\)

IF 2.2 1区 物理与天体物理 Q1 PHYSICS, MATHEMATICAL
Fumio Hiai, Anna Jenčová
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引用次数: 0

Abstract

We study the \(\alpha \)-z-Rényi divergences \(D_{\alpha ,z}(\psi \Vert \varphi )\) where \(\alpha ,z>0\) (\(\alpha \ne 1\)) for normal positive functionals \(\psi ,\varphi \) on general von Neumann algebras, introduced in Kato and Ueda (arXiv:2307.01790) and Kato (arXiv:2311.01748). We prove the variational expressions and the data processing inequality (DPI) for the \(\alpha \)-z-Rényi divergences. We establish the sufficiency theorem for \(D_{\alpha ,z}(\psi \Vert \varphi )\), saying that for \((\alpha ,z)\) inside the DPI bounds, the equality \(D_{\alpha ,z}(\psi \circ \gamma \Vert \varphi \circ \gamma )=D_{\alpha ,z}(\psi \Vert \varphi )<\infty \) in the DPI under a quantum channel (or a normal 2-positive unital map) \(\gamma \) implies the reversibility of \(\gamma \) with respect to \(\psi ,\varphi \). Moreover, we show the monotonicity properties of \(D_{\alpha ,z}(\psi \Vert \varphi )\) in the parameters \(\alpha ,z\) and their limits to the normalized relative entropy as \(\alpha \nearrow 1\) and \(\alpha \searrow 1\).

\von Neumann Algebras 中的(\alpha \)-z-Rényi 分歧:(\\alpha ,z)中的数据处理不等式、可逆性和单调性特性
我们研究了一般冯-诺伊曼代数上的正常正函数(D_{\alpha ,z}(\psi \Vert \varphi )\) where \(\alpha ,z>0\) (\(\alpha \ne 1\)) 的 \(\alpha ,z>0\) (\(\alpha \ne 1\)) 分歧,在 Kato 和 Ueda (arXiv:2307.01790) 和 Kato (arXiv:2311.01748)中引入。我们证明了 \(\alpha \)-z-Rényi分歧的变分表达式和数据处理不等式(DPI)。我们为 \(D_{\alpha ,z}(\psi \Vert \varphi )\) 建立了充分性定理,即对于 DPI 边界内的((\alpha ,z)\)、等式(D_{alpha ,z}(\psi \circ \gamma \Vert \varphi \circ \gamma )=D_{alpha ,z}(\psi \Vert \varphi )<;\在DPI中,在量子通道(或正常的2-positive unital map)下的\(\gamma\)意味着\(\gamma\)相对于\(\psi ,\varphi\)的可逆性。)此外,我们还展示了参数\(\alpha ,z\)中的\(D_{\alpha ,z}(\psi \Vert \varphi )\)的单调性以及它们对归一化相对熵的限制,即\(\alpha \nearrow 1\) 和\(\alpha \searrow 1\).
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来源期刊
Communications in Mathematical Physics
Communications in Mathematical Physics 物理-物理:数学物理
CiteScore
4.70
自引率
8.30%
发文量
226
审稿时长
3-6 weeks
期刊介绍: The mission of Communications in Mathematical Physics is to offer a high forum for works which are motivated by the vision and the challenges of modern physics and which at the same time meet the highest mathematical standards.
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