论文标题
差异动态扩展的独特性和最佳性
Uniqueness and Optimality of Dynamical Extensions of Divergences
论文作者
论文摘要
我们引入了一种用于通道差异和通道相对熵的公理方法，该方法基于超级通道下的三个信息理论公理（即广义数据处理不平等），张量张量下的添加性和归一化，类似于该州域最近给出的方法。我们表明，这些公理足以在通道域中提供足够的结构，从而导致许多适用于所有通道差异的属性。这些包括忠诚，连续性，一种三角形不平等以及最小渠道相对熵之间的界限。此外，我们证明了独特的定理，表明kullback-leibler差异仅具有对经典通道的扩展。对于量子通道，除了最大相对熵之外，这种唯一性无法保持。取而代之的是，我们通过证明它在所有通道相对熵上提供了一个降低至kullback-leibler差异的较低限制，从而证明了Umegaki相对熵的摊销通道扩展的最佳性。我们还介绍了给定的经典状态差异的最大通道扩展并研究其特性。
We introduce an axiomatic approach for channel divergences and channel relative entropies that is based on three information-theoretic axioms of monotonicity under superchannels (i.e. generalized data processing inequality), additivity under tensor products, and normalization, similar to the approach given recently for the state domain. We show that these axioms are sufficient to give enough structure also in the channel domain, leading to numerous properties that are applicable to all channel divergences. These include faithfulness, continuity, a type of triangle inequality, and boundedness between the min and max channel relative entropies. In addition, we prove a uniqueness theorem showing that the Kullback-Leibler divergence has only one extension to classical channels. For quantum channels, with the exception of the max relative entropy, this uniqueness does not hold. Instead we prove the optimality of the amortized channel extension of the Umegaki relative entropy, by showing that it provides a lower bound on all channel relative entropies that reduce to the Kullback-Leibler divergence on classical states. We also introduce the maximal channel extension of a given classical state divergence and study its properties.
