Realigning random states

Source abstract

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 Cd⊗Cd obtained by partial tracing a Haar-distributed random pure state on Cd⊗Cd⊗Cs over an ancilla space Cs. We show that, for large d, typically, the realignment criterion is successful in detecting entanglement if and only if \documentclass[12pt]{minimal}\begin{document}$s \leqslant (\frac{8}{3\pi })^2 d^2$\end{document}s⩽(83π)2d2. In this sense, the realignment criterion is asymptotically weaker than the partial transposition criterion.