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 $\C^d \otimes \C^d$ obtained by partial tracing a Haar-distributed random pure state on $\C^d \otimes \C^d \otimes \C^s$ over an ancilla space $\C^s$. We show that, for large $d$, the realignment criterion typically detects entanglement if and only if $s \leq (8/3\pi)^2 d^2$. In this sense, the realignment criterion is asymptotically weaker than the partial transposition criterion.