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⊗C^d obtained by partial tracing a Haar-distributed random pure state on C^d⊗C^d⊗C^s over an ancilla space C^s. We show that, for large d, typically, the realignment criterion is successful in detecting entanglement if and only if $s⩽{\left(\frac{8}{3\pi }\right)}^2{d}^2$ . In this sense, the realignment criterion is asymptotically weaker than the partial transposition criterion.