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 $m\ge n$, fix an arbitrary state $\omega$ of $N$, and let $E(\omega )$ be the set of all states of $M\otimes N$ that extend $\omega$. The space $E(\omega )$ 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 $E^r(\omega )$ is the set of all states of $E(\omega )$ having rank $\le r$. 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 $E^r(\omega )$. The unique $G^r$-invariant probability measure on $V^r$ promotes to a probability measure $P^{r,\omega }$ on $E^r(\omega )$, and $P^{r,\omega }$ assigns probability 1 to states of rank $r$. The resulting probability space $(E^r(\omega ),P^{r,\omega })$ represents ``choosing a rank $r$ extension of $\omega$ at random". Main result: For every $r=1,2,...,[\rank ω/2]$, states of $(E^r(\omega ),P^{r,\omega })$ are almost surely entangled.

Significant revisions and clarifications, more references. 33 pages

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• Maximal rank of extremal marginal tracial states
• A generic quantum Wielandt's inequality