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

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 operator space 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 ϑ function, as the norm-completion (or stabilisation) of a "naive" generalisation of ϑ. 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 ϑ in the classical case, give several applications, and propose to study the operator spaces associated to channels as "non-commutative graphs", using the language of Hilbert modules.

Lovász theta type norms and operator systems

“…We now see that [3]. Similarly, ϑ cb (S ⊥ G ) =θ(S G ) is the "complete" Lovász number of G introduced in [3].…”

“…Results in [3] imply that S ⊥ G is a kernel in our sense. Below is a direct proof in the language of operator systems, that also characterizes the quotient as the operator system dual of S G .…”

“…We now wish to relate this to a Lovász theta type parameter, which was first introduced in [3]. Again when n = 1 we set ϑ(J ) := ϑ 1 (J ).…”

Zoology of Atlas-Groups: Dessins D’enfants, Finite Geometries and Quantum Commutation

“…The first one consists of computing the Lovasz number of the incidence graph of the geometry in question, e.g., [21], the second one is based on cohomological concepts [22].…”

“…There exist 416 orthogonal bases, that is triples of mutually orthogonal non-isotropic points. The resulting configuration [208 6 , 416 3 ] (5) has been shown to be related to a 3 − (66, 16,21) design used to construct the Suzuki sporadic group Suz [32] (see also Table 12 below).…”