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

math.OACross-listedarXiv:2512.10410Dec 12, 2025

Tensor products of convex sets and the volume of separable states on N qudits

Guillaume Aubrun, Stanisław J. Szarek

This paper deals with estimating the volume of the set of separable mixed quantum states when the dimension of the state space grows to infinity. This has been studied recently for qubits; here we consider larger particles and conclude that, in all cases, the proportion of the states that are separable is superexponentially small in the dimension of the set. We also show that the partial transpose criterion becomes imprecise when the dimension increases, and that the lower bound ${6}^{\ensuremath{-}N∕2}$ on the (Hilbert-Schmidt) inradius of the set of separable states on $N$ qubits obtained recently by Gurvits and Barnum is essentially optimal. We employ standard tools of classical convexity, high-dimensional probability, and geometry of Banach spaces. One relatively nonstandard point is a formal introduction of the concept of projective tensor products of convex bodies, and an initial study of this concept.This paper deals with estimating the volume of the set of separable mixed quantum states when the dimension of the state space grows to infinity. This has been studied recently for qubits; here we consider larger particles and conclude that, in all cases, the proportion of the states that are separable is superexponentially small in the dimension of the set. We also show that the partial transpose criterion becomes imprecise when the dimension increases, and that the lower bound ${6}^{\ensuremath{-}N∕2}$ on the (Hilbert-Schmidt) inradius of the set of separable states on $N$ qubits obtained recently by Gurvits and Barnum is essentially optimal. We employ standard tools of classical convexity, high-dimensional probability, and geometry of Banach spaces. One relatively nonstandard point is a formal introduction of the concept of projective tensor products of convex bodies, and an initial study of this concept.

arXiv:quant-ph/0503221Physical Review AFeb 14, 200674 citations

Classification of Separable States for Tripartite Qudits via Bell Inequalities

Chen Xie, Hui Zhao, Zhi‐Xi Wang

International Journal of Theoretical PhysicsNov 23, 20121 citation

Entanglement in C

^*

-algebras: tensor products of state spaces

We analyze the Namioka-Phelps minimal and maximal tensor products of compact convex sets which arise as the state spaces of unital C

^*

-algebras. Relatedly, we study entanglement in (infinite dimensional) C

^*

-algebras. While the minimal Namioka-Phelps tensor product of the state spaces of two C

^*

-algebras is the well-known set of separable (= un-entangled) states on the (minimal) tensor product of the C

^*

-algebras, we also describe the more elusive maximal Namioka-Phelps tensor product of state spaces of C

^*

-algebras. We show that the minimal and maximal tensor products of state spaces of C

^*

-algebras agree precisely when one of the two C

^*

-algebras is commutative, which confirms Barker's conjecture in the case where the compact convex sets are state paces of C

^*

-algebras. Further, the Namioka-Phelps tensor product of the trace simplexes of two or more unital C

^*

-algebras is shown to be the trace simplex of the (minimal or maximal) tensor product of the C

^*

-algebras. This enables a systematic way of determining the trace simplex of a tensor product of C

^*

-algebras.

N

qubits obtained recently by Gurvits and Barnum is essentially optimal. We employ standard tools of classical convexity, high-dimensional probability, and geometry of Banach spaces. One relatively nonstandard point is a formal introduction of the concept of projective tensor products of convex bodies, and an initial study of this concept.