The Probability of Entanglement

Abstract

We show that states on tensor products of matrix algebras whose ranks are relatively small are almost surely entangled, but that states of maximum rank are not. More precisely, let $M = M_m(\mathbb{C})$ and ${N=M_n(\mathbb C)}$ be full matrix algebras with m ≥  n, fix an arbitrary state ω of N, and let E(ω) be the set of all states of ${M\otimes N}$ that extend ω. The space E(ω) contains states of rank r for every r = 1, 2, . . . , m · rank ω, and it has a filtration into compact subspaces

$$E^1(\omega)\subseteq E^2(\omega)\subseteq \cdots\subseteq E^{m\cdot{\rm rank}\,\omega}=E(\omega),$$

where E ^r(ω) is the set of all states of E(ω) having rank  ≤  r.

We show first that for every r, there is a real-analytic manifold V ^r, homogeneous under a transitive action of a compact group G ^r, which parameterizes E ^r(ω). The unique G ^r-invariant probability measure on V ^r promotes to a probability measure P ^r,ω on E ^r(ω), and P ^r,ω assigns probability 1 to states of rank r. The resulting probability space (E ^r(ω),P ^r,ω) represents “choosing a rank r extension of ω at random”.

Main result: For every r = 1, 2, . . . , [rank ω/2], states of (E ^r(ω),P ^r,ω) are almost surely entangled.