Zero-error communication via quantum channels and a quantum Lovász script v sign-function

Abstract

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 linear space operators as the quantum generalisation of the adjacency matrix, in terms of which the plain, quantum and entanglement-assisted capacity can be formulated, and for which we show some new basic properties. Most importantly, we define a quantum version of Lovász' famous script v sign function, as the norm-completion (or stabilisation) of a "naive" generalisation of script v sign. We go on to show that this function upper bounds the number of entanglement-assisted zero-error messages, that it is given by a semidefinite programme, whose dual we write down explicitly, and that it is multiplicative with respect to the natural (strong) graph product. We explore various other properties of the new quantity, which reduces to Lovász' original script v sign in the classical case, give several applications, and propose to study the linear spaces of operators associated to channels as "non-commutative graphs", using the language of operator systems and Hilbert modules. © 2011 IEEE.