Quantum communication relies on the existence of high quality quantum channels to exchange information. In practice, however, all communication links are affected by noise from the environment. Here we investigate the ability of quantum channels to perform quantum communication tasks by restricting the participants to use only local operations and one-way classical communication (one-way LOCC) along with the available quantum channel. In particular, a channel can be used to distill a highly entangled state between two parties, which further enables quantum or private communication. In this work, we invoke the framework of superchannels to study the distillation of a resourceful quantum state, such as a maximally entangled state or a private state, using multiple instances of a point-to-point quantum channel. We use the idea of $k$-extendibility to obtain a semidefinite relaxation of the set of one-way LOCC superchannels and define a class of entanglement measures for quantum channels that decrease monotonically under such superchannels; therefore these measures, dubbed collectively the ``unextendible entanglement of a channel'', yield upper bounds on several communication-theoretic quantities of interest in the regimes of resource distillation and zero error. We then generalize the formalism of $k$-extendibility to bipartite superchannels, thus obtaining functions that are monotone under two-extendible superchannels. This allows us to analyze probabilistic distillation of ebits or secret key bits from a bipartite state when using a resourceful quantum channel. Moreover, we propose semidefinite programs to evaluate several of these quantities, providing a computationally feasible method of comparison between quantum channels for resource distillation.

翻译：量子通信依赖于高质量量子信道的存在以交换信息。然而在实际中，所有通信链路都会受到环境噪声的影响。本文通过限制参与者仅使用局部操作与单向经典通信（单向LOCC）以及可用量子信道，研究量子信道执行量子通信任务的能力。特别地，信道可用于在两个参与方之间蒸馏出高度纠缠态，从而进一步实现量子通信或保密通信。本工作中，我们引入超信道框架来研究使用多个点对点量子信道实例蒸馏资源型量子态（如最大纠缠态或保密态）的问题。我们利用$k$-可扩展性概念获得单向LOCC超信道集合的半定松弛，并定义了一类在超信道作用下单调递减的量子信道纠缠度量；这些被统称为“信道不可扩展纠缠”的度量，为资源蒸馏和零错误场景下的若干通信理论量提供了上界。随后我们将$k$-可扩展性形式推广至二分超信道，从而得到在二可扩展超信道下单调的函数。这使得我们能够分析在使用资源型量子信道时，从二分态概率性蒸馏纠缠比特或保密密钥比特的过程。此外，我们提出了评估其中若干量的半定规划，为量子信道在资源蒸馏方面的比较提供了计算可行的方法。