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#### Functional AnalysisInformation TheoryConvex and Discrete Geometry mathscidoc:1805.12001

Comm. Pure Appl. Math., 67, (1), 129-171, 2014.1
For a random quantum state on $\cH=\C^d \otimes \C^d$ obtained by partial tracing a random pure state on $\cH \otimes \C^s$, 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 $\e>0$ and for $d$ large enough, such a random state is entangled with very large probability when $s \leq (1-\e)s_0$, and separable with very large probability when $s \geq (1+\e) s_0$. 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 $\C^{d_1} \otimes \C^{d_2}$, ${d_1} \neq {d_2}$. 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.
Quantum entanglement, random matrices, concentration of measures, convex geometry, quantum information theory
• Our results in this paper not only solved a long-standing question regarding the ubiquity of quantum entanglement (i.e., Einstein's 'spooky action at a distance'), but also provided additional parameters to quantum information theory. This paper received extensive public attention. Several non-mathematical articles, such as "Einstein's 'spooky action' common in large quantum systems", "Quantum entanglement isn't only spooky, you can't avoid it" and "Quantum Entanglement Common In Large System", were written to explain our results; these articles were widely spread over the internet with more than 369000 Google search items as of 09/2013.
@inproceedings{guillaume2014entanglement,
title={Entanglement Thresholds for Random Induced States},
author={Guillaume Aubrun, Stanislaw J. Szarek, and Deping Ye},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180529231404439593089},
booktitle={Comm. Pure Appl. Math.},
volume={67},
number={1},
pages={129-171},
year={2014},
}

Guillaume Aubrun, Stanislaw J. Szarek, and Deping Ye. Entanglement Thresholds for Random Induced States. 2014. Vol. 67. In Comm. Pure Appl. Math.. pp.129-171. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180529231404439593089.