Entanglement in C$^*$-algebras: tensor products of state spaces
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910129762336768 |
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| author | Musat, Magdalena Rørdam, Mikael |
| author_facet | Musat, Magdalena Rørdam, Mikael |
| contents | 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). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_10410 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Entanglement in C$^*$-algebras: tensor products of state spaces Musat, Magdalena Rørdam, Mikael Operator Algebras Functional Analysis Quantum Physics 46L10, 81P45, 47L90 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). |
| title | Entanglement in C$^*$-algebras: tensor products of state spaces |
| topic | Operator Algebras Functional Analysis Quantum Physics 46L10, 81P45, 47L90 |
| url | https://arxiv.org/abs/2512.10410 |