AI 中文总结
研究半有限冯·诺依曼代数中最大相对熵的平滑指数，通过开发算子代数替代物给出精确公式。还用其制定催化量子信息解耦，证明相关引理，得出解耦可靠性指数由与有限维理论相同的公式描述。
AI 中文摘要
我们研究了半有限冯·诺依曼代数中最大相对熵的平滑指数。主要结果给出了此情形下的精确指数公式。证明过程开发了算子代数替代物来取代有限维论证中依赖维度的工具。这些要素表明平滑指数由基础冯·诺依曼代数结构决定，而非矩阵维度估计。作为应用，我们用半有限冯·诺依曼代数参考系统制定了催化量子信息解耦。我们证明了冯·诺依曼代数的一个内在层蛋糕引理，它消除了有限维证明中的可数谱假设并得出相应的半有限估计。因此，解耦可靠性指数由与有限维理论中相同的夹逼雷尼互信息公式描述。
英文摘要
We establish the exact exponent for smoothing the max-relative entropy on arbitrary von Neumann algebras. The proof first develops the required large-deviation and smoothing estimates in the semifinite setting and then passes to general von Neumann algebras through Haagerup-Junge-Xu reduction. We next study catalytic quantum information decoupling when the reference system is described by an arbitrary von Neumann algebra. A key ingredient is an operator layer-cake formula valid on arbitrary von Neumann algebras. This formula leads to a convex-split estimate for normal states with a general von Neumann algebraic reference system. Combining this estimate with the smoothing exponent result, we obtain lower and upper bounds on the catalytic-decoupling reliability function. These bounds coincide below the corresponding critical rate, yielding the exact decoupling reliability function in that regime.
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