论文标题
von Neumann代数中的渐近电流定理
Asymptotic Equipartition Theorems in von Neumann algebras
论文作者
论文摘要
信息理论中的渐近电气特性（AEP）确定独立和相同分布的（i.i.d.）状态的行为与统一状态相似。特别是，在适当的平滑状态下，对于这种状态，最小和最大相对熵渐近地与相对熵一致。在本文中，我们将几种此类等级特性推广到一般的冯·诺伊曼代数上。 首先，我们表明I.I.D.的光滑最大相对熵von Neumann代数上的状态具有量子相对熵给出的渐近率。实际上，我们的AEP不仅适用于州，而且适用于具有适当限制的量子通道。此外，超越了I.I.D.假设，我们表明，对于由量子通道的顺序过程产生的状态，光滑的最大相对熵可以由适当的通道相对熵的总和上限。 我们的主要技术贡献是扩展到通用的von Neumann代数的背景，这是量子通道的链条规则，以及带有替代通道的通道相对熵的添加性结果。
The Asymptotic Equipartition Property (AEP) in information theory establishes that independent and identically distributed (i.i.d.) states behave in a way that is similar to uniform states. In particular, with appropriate smoothing, for such states both the min and the max relative entropy asymptotically coincide with the relative entropy. In this paper, we generalize several such equipartition properties to states on general von Neumann algebras. First, we show that the smooth max relative entropy of i.i.d. states on a von Neumann algebra has an asymptotic rate given by the quantum relative entropy. In fact, our AEP not only applies to states, but also to quantum channels with appropriate restrictions. In addition, going beyond the i.i.d. assumption, we show that for states that are produced by a sequential process of quantum channels, the smooth max relative entropy can be upper bounded by the sum of appropriate channel relative entropies. Our main technical contributions are to extend to the context of general von Neumann algebras a chain rule for quantum channels, as well as an additivity result for the channel relative entropy with a replacer channel.
