Entanglement in C$^*$-algebras: tensor products of state spaces

TL;DR

This paper investigates tensor products of state spaces in C$^*$-algebras, revealing conditions for entanglement and confirming a longstanding conjecture about the structure of these tensor products.

Contribution

It establishes the equivalence of minimal and maximal tensor products of state spaces when one algebra is commutative, confirming Barker's conjecture in this context.

Findings

Minimal tensor product corresponds to separable states.

Maximal and minimal tensor products agree iff one algebra is commutative.

Trace simplexes of tensor products relate to the Poulsen simplex.

Abstract

We analyze the Namioka-Phelps minimal and maximal tensor products of compact convex sets arising as state spaces of C$^*$-algebras, and, relatedly, study entanglement in (infinite dimensional) C$^*$-algebras. The minimal Namioka-Phelps tensor product of the state spaces of two C$^*$-algebras is shown to correspond to the set of separable (= un-entangled) states on the tensor product of the C$^*$-algebras. We show that these maximal and minimal tensor product of the state spaces agree precisely when one of the two C$^*$-algebras is commutative. This confirms an old conjecture by Barker in the case where the compact convex sets are state spaces of C$^*$-algebras. The Namioka-Phelps tensor product of the trace simplexes of two C$^*$-algebras is shown always to be the trace simplex of the tensor product of the C$^*$-algebras. This can be used, for example, to show that the trace simplex of (any) tensor product of C$^*$-algebras is the Poulsen simplex if and only if the trace simplex of each of the C$^*$-algebras is the Poulsen simplex or trivial (and not all trivial).