Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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
$\operatorname{C}(\mathbb{M}_{\mathbb{P},\mathbb{O}})$.

13
The categories of this paper are: quant-ph
3
The categories of this paper are: physics
0
The categories of this paper are: hep-ph
12
The categories of this paper are: nucl-th
30
The categories of this paper are: q-alg
27
The categories of this paper are: comp-gas
21
The categories of this paper are: adap-org
35
The categories of this paper are: mtrl-th
37
The categories of this paper are: supr-con
22
The categories of this paper are: chao-dyn
28
The categories of this paper are: alg-geom
4
The categories of this paper are: cond-mat
32
The categories of this paper are: atom-ph
11
The categories of this paper are: hep-lat
25
The categories of this paper are: patt-sol
18
The categories of this paper are: econ