The probability of entanglement

Abstract

We show that states on tensor products of matrix algebras whose ranks are relatively small are {\em 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 $m\geq n$, 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 $r=1,2,...,m\cdot\rankω$, and it has a filtration into compact subspaces $$ E^1(ω)\subseteq E^2(ω)\subseteq ...\subseteq E^{m\cdot\rankω}=E(ω), $$ where $E^r(ω)$ is the set of all states of $E(ω)$ having rank $\leq r$. 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.