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. We show that a threshold occurs when the environment dimension s is of order roughly d^3. More precisely, when s \leq cd^3, such a random state is entangled with very large probability, while when s \geq Cd^3 \log^2 d, it is separable with very large probability (here C,c>0 are appropriate effectively computable universal constants). Our proofs rely on random matrices, classical convexity, high-dimensional probability and geometry of Banach spaces.
aubrun (arrobas) math. univ-lyon1. fr