We give a useful new characterization of the set of all completely positive, trace-preserving maps Φ : M 2 → M 2 from which one can easily check any trace-preserving map for complete positivity. We also determine explicitly all extreme points of this set, and give a useful parameterization after reduction to a certain canonical form. This allows a detailed examination of an important class of non-unital extreme points that can be characterized as having exactly two images on the Bloch sphere.We also discuss a number of related issues about the images and the geometry of the set of stochastic maps, and show that any stochastic map on M 2 can be written as a convex combination of two "generalized" extreme points.

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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 −N/2 on the ͑Hilbert-Schmidt͒ 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.