Entanglement thresholds for random induced states

TL;DR

This paper identifies a sharp threshold for the entanglement of random induced quantum states, showing when such states are typically entangled or separable based on the environment dimension, with implications for multipartite systems.

Contribution

The paper establishes a precise entanglement threshold for random induced states and extends the analysis to multipartite and unbalanced systems using advanced mathematical tools.

Findings

Existence of a sharp entanglement threshold s_0(d) of order roughly d^3.

States are entangled with high probability when s < (1-a)s_0, separable when s > (1+a)s_0.

For N-particle systems, a threshold k_0 ~ N/5 determines typical entanglement sharing.

Abstract

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.