Entanglement thresholds for random induced states

For a random quantum state on $H=C^d \otimes C^d$ obtained by partial tracing a random pure state on $H \otimes C^s$, we consider the whether it is typically separable or typically entangled. For this problem, we show the existence of a sharp threshold $s_0=s_0(d)$ of order roughly $d^3$. More precisely, for any $a > 0$ and for d large enough, such a random state is entangled with very large probability when $s < (1-a)s_0$, and separable with very large probability when $s > (1+a)s_0$. One consequence of this result is as follows: for a system of N identical particles in a random pure state, there is a threshold $k_0 = k_0(N) \sim N/5$ such that two subsystems of k particles each typically share entanglement if $k > k_0$, and typically do not share entanglement if $k < k_0$. Our methods work also for multipartite systems and for "unbalanced" systems such as $C^{d} \otimes C^{d'}$, $d \neq d'$. The arguments rely on random matrices, classical convexity, high-dimensional probability and geometry of Banach spaces; some of the auxiliary results may be of reference value. A high-level non-technical overview of the results of this paper and of a related article arXiv:1011.0275 can be found in arXiv:1112.4582.