Quantum correlations on quantum spaces

Abstract

For given quantum (non-commutative) spaces $\mathbb{P}$ and $\mathbb{O}$ we study the quantum space of maps ${\mathbb{M}}_{\mathbb{P},\mathbb{O}}$ from $\mathbb{P}$ to $\mathbb{O}$. In case of finite quantum spaces these objects turn out to be behind a large class of maps which generalize the classical $\mathrm{qc}$-correlations known from quantum information theory to the setting of quantum input and output sets. We prove a number of important functorial properties of the mapping $(\mathbb{P},\mathbb{O})\mapsto {\mathbb{M}}_{\mathbb{P},\mathbb{O}}$ and use them to study various operator algebraic properties of the ${\mathrm{C}}^{\ast }$-algebras $C({\mathbb{M}}_{\mathbb{P},\mathbb{O}})$ such as the lifting property and residual finite dimensionality. Inside $C({\mathbb{M}}_{\mathbb{P},\mathbb{O}})$ we construct a universal operator system ${\mathbb{S}}_{\mathbb{P},\mathbb{O}}$ related to $\mathbb{P}$ and $\mathbb{O}$ and show, among other things, that the embedding ${\mathbb{S}}_{\mathbb{P},\mathbb{O}}\subset C({\mathbb{M}}_{\mathbb{P},\mathbb{O}})$ is hyperrigid, $C({\mathbb{M}}_{\mathbb{P},\mathbb{O}})$ is the ${\mathrm{C}}^{\ast }$-envelope of ${\mathbb{S}}_{\mathbb{P},\mathbb{O}}$ and that a large class of non-signalling correlations on the quantum sets $\mathbb{P}$ and $\mathbb{O}$ arise from states on $C({\mathbb{M}}_{\mathbb{P},\mathbb{O}}){\otimes }_{\max }C({\mathbb{M}}_{\mathbb{P},\mathbb{O}})$ as well as states on the commuting tensor product ${\mathbb{S}}_{\mathbb{P},\mathbb{O}}{\otimes }_{\mathrm{c}}{\mathbb{S}}_{\mathbb{P},\mathbb{O}}$. Finally we introduce and study the notion of a synchronous correlation with quantum input and output sets, prove several characterizations of such correlations and their relation to traces on $C({\mathbb{M}}_{\mathbb{P},\mathbb{O}})$.

Some arguments were shortened and streamlined, some less interesting parts were removed