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 < (8/3π)^2 d^2 . In this sense, the realignment criterion is asymptotically weaker than the partial transposition criterion.
E-mail :
aubrun (arrobas) math. univ-lyon1. fr