The probability of entanglement

TL;DR

This paper demonstrates that in tensor products of matrix algebras, states with small rank are almost surely entangled, while maximum rank states are typically not, revealing probabilistic entanglement properties.

Contribution

It introduces a probabilistic framework for analyzing entanglement in states of tensor product matrix algebras based on rank and extends previous understanding of entanglement distribution.

Findings

States of low rank are almost surely entangled.

States of maximum rank are not entangled.

Probabilistic measures favor entanglement in low-rank states.

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 $\omega$ 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 $r=1,2,...,m\cdot\rank\omega$, and it has a filtration into compact subspaces $$ E^1(\omega)\subseteq E^2(\omega)\subseteq ...\subseteq E^{m\cdot\rank\omega}=E(\omega), $$ where $E^r(\omega)$ is the set of all states of $E(\omega)$ 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 $E^r(\omega)$. 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 $r=1,2,...,[\rank \omega/2]$, states of $(E^r(\omega),P^{r,\omega})$ are almost surely entangled.