Zero-Error Communication via Quantum Channels, Noncommutative Graphs, and a Quantum Lovasz Number

We study the quantum channel version of Shannon's zero-error capacity problem. Motivated by recent progress on this question, we propose to consider a certain subspace of operators (so-called operator systems) as the quantum generalisation of the adjacency matrix, in terms of which the zero-error capacity of a quantum channel, as well as the quantum and entanglement-assisted zero-error capacities can be formulated, and for which we show some new basic properties. Most importantly, we dene a quantum version of Lovasz' famous # function on general operator systems, as the norm- completion (or stabilisation) of a naive" generalisation of #. We go on to show that this function upper bounds the number of entanglement-assisted zero-error messages, that it is given by a semidenite programme, whose dual we write down explicitly, and that it is multiplicative with respect to the tensor product of operator systems (corresponding to the tensor product of channels). We explore various other properties of the new quantity, which reduces to Lovasz' original # in the classical case, give several applications, and propose to study the operator sys- tems associated to channels as non-commutative graphs", using the language of Hilbert modules.