This is an announcement for the paper "Entanglement thresholds for random induced states" by Guillaume Aubrun, Stanislaw J. Szarek and Deping Ye.
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. 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. Our methods work also for multipartite systems and for "unbalanced" systems such as $C^{d} \otimes C^{d'}$, $d \neq d' $.
Archive classification: quant-ph math.FA math.PR
Report Number: Mittag-Leffler-2010fall
Remarks: 29 pages
Submitted from: szarek@cwru.edu
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1106.2264
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