Realigning random states

We study how the realignment criterion (also called computable cross-norm criterion) succeeds asymptotically in detecting whether random states are separable or entangled. We consider random states on ${\text{C}}^d\otimes {\text{C}}^d$ obtained by partial tracing a Haar-distributed random pure state on $\C^d \otimes \C^d \otimes \C^s$overanancillaspace$\C^s$. We show that, for large $d$, the realignment criterion typically detects entanglement if and only if $s\le (8\mathrm{/}3\pi {)}^2{d}^2$. In this sense, the realignment criterion is asymptotically weaker than the partial transposition criterion.