Entanglement in C${}^{\ast }$-algebras: tensor products of state spaces

Abstract

We analyze the Namioka-Phelps minimal and maximal tensor products of compact convex sets which arise as the state spaces of unital C${}^{\ast }$-algebras. Relatedly, we study entanglement in (infinite dimensional) C${}^{\ast }$-algebras. While the minimal Namioka-Phelps tensor product of the state spaces of two C${}^{\ast }$-algebras is the well-known set of separable (= un-entangled) states on the (minimal) tensor product of the C${}^{\ast }$-algebras, we also describe the more elusive maximal Namioka-Phelps tensor product of state spaces of C${}^{\ast }$-algebras. We show that the minimal and maximal tensor products of state spaces of C${}^{\ast }$-algebras agree precisely when one of the two C${}^{\ast }$-algebras is commutative, which confirms Barker's conjecture in the case where the compact convex sets are state paces of C${}^{\ast }$-algebras. Further, the Namioka-Phelps tensor product of the trace simplexes of two or more unital C${}^{\ast }$-algebras is shown to be the trace simplex of the (minimal or maximal) tensor product of the C${}^{\ast }$-algebras. This enables a systematic way of determining the trace simplex of a tensor product of C${}^{\ast }$-algebras.

27 pages. This version has undergone significant revisions, including correcting a wrong statement about tensor products of Poulsen simplexes