Entanglement Thresholds for Random Induced States

For a random quantum state on H = C d ⊗ C d obtained by partial tracing a random pure state on H ⊗ C s, we consider the question 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 ε > 0 and for d large enough, such a random state is entangled with very large probability when s ≤ ( 1 − ε ) s 0, and separable with very large probability when s ≥ ( 1 + ε ) 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 ) ~ N / 5 such that two subsystems of k particles each typically share entanglement if k > k0, and typically do not share entanglement if k < k0. Our methods also work for multipartite systems and for “unbalanced” systems such as C d 1 ⊗ C d 2, d 1 ≠ d 2. 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. © 2013 Wiley Periodicals, Inc.