Achieving the Han-Kobayashi inner bound for the quantum interference channel

We construct an encoding and decoding scheme achieving the Chong-Motani-Garg inner bound [1] for a two sender two receiver interference channel with classical input and quantum output. This automatically gives a similar inner bound for sending classical information through an interference channel with quantum inputs and outputs without entanglement assistance. Our result matches the best known inner bound for the interference channel in the classical setting. Achieving the Chong-Motani-Garg inner bound, which is known to be equivalent to the Han-Kobayashi inner bound [3], answers an open question raised recently by Fawzi et al. [4]. Our encoding strategy is the standard random encoding strategy. Our decoding strategy is a sequential strategy where a receiver loops through all candidate messages trying to project the received state onto a `typical' subspace for the candidate message under consideration, stopping if the projection succeeds for a message, which is then declared as the guess of the receiver for the sent message. On the way to our main result, we show that random encoding and sequential decoding strategies suffice to achieve rates up to the mutual information for a single sender single receiver channel, and the standard inner bound for a two sender single receiver multiple access channel, for channels with classical input and quantum output. Besides conceptual simplicity, a sequential decoding strategy is space efficient, and may have additional efficiency advantages in some settings. We prove our inner bounds using two new technical tools - a non-commutative union bound to analyse the decoding error probability, and a geometric notion of approximate interesection of two conditionally typical subspaces.