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- W4285178249 abstract "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 $textrm {qc}$-correlations known from quantum information theory to the setting of quantum input and output sets. We prove various operator algebraic properties of the ${textrm C}^ast$-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 and has another interesting property, which we call the strong extension property. Furthermore, ${operatorname {C}}({mathbb {M}}_{{mathbb {P}},{mathbb {O}}})$ is the ${textrm C}^ast$-envelope of ${mathbb {S}}_{{mathbb {P}},{mathbb {O}}}$ and a large class of non-signalling correlations on the quantum sets ${mathbb {P}}$ and ${mathbb {O}}$ arise from states on ${operatorname {C}}({mathbb {M}}_{{mathbb {P}},{mathbb {O}}})otimes _{text {tiny {textrm {max}}}}{operatorname {C}}({mathbb {M}}_{{mathbb {P}},{mathbb {O}}})$ as well as states on the commuting tensor product ${mathbb {S}}_{{mathbb {P}},{mathbb {O}}}otimes _{text {tiny {textrm {c}}}}{mathbb {S}}_{{mathbb {P}},{mathbb {O}}}$. Finally, we introduce and study the notion of a synchronous correlation with quantum input and output sets and prove several characterizations of such correlations and their relation to traces on ${operatorname {C}}({mathbb {M}}_{{mathbb {P}},{mathbb {O}}})$." @default.
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- W4285178249 date "2022-07-14" @default.
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- W4285178249 title "Quantum Correlations on Quantum Spaces" @default.
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- W4285178249 doi "https://doi.org/10.1093/imrn/rnac139" @default.
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