Zeroerror communication via quantum channels, noncommutative graphs and a quantum Lovasz theta function
Abstract
We study the quantum channel version of Shannon's zeroerror capacity problem. Motivated by recent progress on this question, we propose to consider a certain operator space as the quantum generalisation of the adjacency matrix, in terms of which the plain, quantum and entanglementassisted capacity can be formulated, and for which we show some new basic properties. Most importantly, we define a quantum version of Lovasz' famous theta function, as the normcompletion (or stabilisation) of a "naive" generalisation of theta. We go on to show that this function upper bounds the number of entanglementassisted zeroerror 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 Lovasz' original theta in the classical case, give several applications, and propose to study the operator spaces associated to channels as "noncommutative graphs", using the language of Hilbert modules.
 Publication:

arXiv eprints
 Pub Date:
 February 2010
 arXiv:
 arXiv:1002.2514
 Bibcode:
 2010arXiv1002.2514D
 Keywords:

 Quantum Physics;
 Computer Science  Information Theory;
 Mathematics  Operator Algebras
 EPrint:
 24 pages: v2 has added discussion and more details in section 7.