Quantum correlations on quantum spaces.

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 $\mathrm{C}^*$-algebras $\operatorname{C}(\mathbb{M}_{\mathbb{P},\mathbb{O}})$ such as the lifting property and residual finite dimensionality. Inside $\operatorname{C}(\mathbb{M}_{\mathbb{P},\mathbb{O}})$ 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 $\mathbb{S}_{\mathbb{P},\mathbb{O}}\subset\operatorname{C}(\mathbb{M}_{\mathbb{P},\mathbb{O}})$ is hyperrigid, $\operatorname{C}(\mathbb{M}_{\mathbb{P},\mathbb{O}})$ is the $\mathrm{C}^*$-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 $\operatorname{C}(\mathbb{M}_{\mathbb{P},\mathbb{O}})\otimes_{\rm{max}}\operatorname{C}(\mathbb{M}_{\mathbb{P},\mathbb{O}})$ as well as states on the commuting tensor product $\mathbb{S}_{\mathbb{P},\mathbb{O}}\otimes_{\rm{c}}\mathbb{S}_{\mathbb{P},\mathbb{O}}$. 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 $\operatorname{C}(\mathbb{M}_{\mathbb{P},\mathbb{O}})$.