Abstract

We consider the problem of transmitting classical and quantum information reliably over an entanglement-assisted quantum channel (EACQ coding). Our main result is a capacity theorem that gives a three-dimensional region containing all achievable rates. Points in the capacity region are rate triples, consisting of the classical communication rate, the quantum communication rate, and the entanglement consumption rate of a particular coding scheme. The crucial protocol in achieving the boundary points of the capacity region is a protocol that we name the classically-enhanced father protocol. The classically-enhanced father protocol is more general than any other protocol in the family tree of quantum Shannon theoretic protocols, in the sense that several previously known quantum protocols are now child protocols of it. The classically-enhanced father protocol also shows an improvement over a time-sharing strategy for the case of a qubit dephasing channel—this result justifies the need for simultaneous coding of classical and quantum information over an entanglement-assisted quantum channel. Our capacity theorem is of a multi-letter nature (requiring a limit over many uses of the channel), but it reduces to a single-letter expression for at least three channels: the completely depolarizing channel, the quantum erasure channel, and the qubit dephasing channel. We also show that isometric encoding is optimal for entanglement-assisted quantum channel.