Volume of the set of separable states

TLDR

The authors introduce a natural measure on N‑dimensional density matrices and investigate the probability that a randomly chosen state is separable. They derive analytical bounds for this probability, perform numerical calculations for N=4 and N=6, and analyze separability under fixed purity conditions. The bounds and calculations show P≈0.632 for N=4, P≈0.384 for N=6, that P decreases exponentially with N, and that low‑purity states are necessarily separable, revealing a duality between purity and entanglement.

Abstract

A natural measure in the space of density matrices describing N-dimensional quantum systems is proposed. We study the probability P that a quantum state chosen randomly with respect to the natural measure is not entangled (is separable). We find analytical lower and upper bounds for this quantity. Numerical calculations give P = 0.632 for N=4 and P=0.384 for N=6, and indicate that P decreases exponentially with N. Analysis of a conditional measure of separability under the condition of fixed purity shows a clear dualism between purity and separability: entanglement is typical for pure states, while separability is connected with quantum mixtures. In particular, states of sufficiently low purity are necessarily separable.