冯·诺依曼代数上夹式和α-z Rényi散度的固定射线护航表示
Fixed-ray escort representations of sandwiched and $α$--$z$ Rényi divergences on von Neumann algebras
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中文总结 AI 辅助
该研究在冯·诺依曼代数框架下，利用Haagerup非交换L^p空间与插值方法，将两类Rényi散度表示为普通相对熵的平均值，导出相关性质并应用于单轮检验等问题。
中文摘要 AI 辅助
我们将夹式Rényi散度和α-z Rényi散度表示为普通相对熵的平均值。α-z Rényi散度可沿射线z=cα对规范族固定射线护航态的相对熵积分得到，我们利用Haagerup非交换L^p空间与插值方法，证明了任意冯·诺依曼代数上正规态的该表示式，公式对所有z>0成立：当0<α<1时，要求第一态的支撑包含于参考态的支撑；当α>1时，只要散度有限即成立。当下阶支撑条件不满足时，我们确定了精确的固定射线支撑边界项。该表示式导出单调护航轮廓与凸序势，我们用它们重述了单轮检验逆、精确夹式强逆指数及功提取可靠性为带符号面积或过零陈述，并讨论了受限双参数对转换率。
英文摘要
We represent sandwiched and $α$-$z$ Rényi divergences as averages of ordinary relative entropy. The $α$-$z$ Rényi divergence is shown to be an integral over the relative entropy of a canonical family of fixed-ray escort states along the ray $z=cα$. We prove this representation for normal states on an arbitrary von Neumann algebra, using Haagerup non-commutative $L^p$ spaces and interpolation. The formula holds for every $z>0$: for $0<α<1$ it holds when the support of the first state is contained in that of the reference state, and for $α>1$ it holds whenever the divergence is finite. When the lower-order support condition fails, we identify the exact fixed-ray support-boundary term. The representation yields a monotone escort profile and a convex order potential. We use these to reformulate one-shot testing converses, exact sandwiched strong-converse exponents, and work-extraction reliability as signed-area or level-crossing statements, and discuss a restricted two-parameter pair-conversion rate.
