The state space of a pair of spin-\(1/2\) particles. (Q1805700)

Let \(H_2\) be a two-dimensional complex Hilbert space and let \(H_2\otimes H_2\) be the tensor product of two copies of \(H_2\). The convex set \(S_2\) of positive operators on \(H_2\otimes H_2\) of trace 1 represents the state space of a pair of spin-\(\tfrac 12\) particles. In this paper, the author makes a detailed study of the set \(S_2\). His techniques involve the use of the co-ordinates of elements of \(S_2\) with respect to the tensor products of orthonormal spin bases for the two \(W^*\)-algebras \(B(H_2)\) and \(B(H_2)\). In particular he considers extreme points of \(S_2\), extreme points of the convex hull of all pure tensors of one-dimensional projections on \(H_2\) and the orbits of extreme points of \(S_2\) under the action of the group of unitary operators on \(H_2\).