One shot entanglement assisted classical and quantum communication over noisy quantum channels: A hypothesis testing and convex split approach

Capacity of a quantum channel characterizes the limits of reliable communication through a noisy quantum channel. This fundamental information theoretic question is very well studied specially in the setting of many independent uses of the channel. An important scenario, both from practical and conceptual point of view, is when the channel can be used only once. This is known as the one-shot channel coding problem. We provide a tight characterization of the one-shot entanglement assisted classical capacity of a quantum channel. We arrive at our result by introducing a simple decoding technique which we refer to as position-based decoding. We also consider two other important quantum network scenarios: quantum channel with a jammer and quantum broadcast channel. For these problems, we use the recently introduced convex split technique [Anshu, Devabathini and Jain 2014] in addition to position based decoding. Our approach exhibits that the simultaneous use of these two techniques provides a uniform and conceptually simple framework for designing communication protocols for quantum networks.