分歧的动力扩张的唯一性和最优性
Uniqueness and Optimality of Dynamical Extensions of Divergences
摘要：
我们引入了一个公理化的方法，通道发散和通道相对熵，是基于三个信息理论公理的单调性下的超级通道（即广义数据处理不等式），张量积下的可加性，和归一化，类似的方法最近给出的状态域。我们表明，这些公理是足够的，以提供足够的结构，也在通道域，导致许多属性，适用于所有通道的分歧。这些包括忠实性，连续性，一种三角不等式，以及最小和最大通道相对熵之间的有界性。此外，我们证明了一个唯一性定理，表明Kullback-Leibler发散只有一个扩展到经典通道。对于量子信道，除了最大相对熵之外，这种唯一性不成立。相反，我们证明了最优的摊销通道扩展的Umegaki相对熵，通过显示，它提供了一个下界的所有通道相对熵，减少到经典状态的Kullback-Leibler分歧。我们还引入了一个给定的经典态发散的最大通道扩张，并研究了它的性质。
英文摘要：
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.
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