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Entanglement thresholds for random induced states. (English) Zbl 1296.81014
In this work the authors consider the question whether a state obtained by partial tracing a random pure state is typically separable or typically entangled. A rigorous proof is presented in terms of asymptotic geometric analysis. The main result states the existence of effectively computable constants $$C, c>0$$ and a function $$s_0(d)$$ satisfying certain conditions such that, from the text, “if $$\rho$$ is a random state on $$\mathbb{C}^d\otimes\mathbb{C}^d$$ distributed according to the measure $$\mu_{d^2,s}$$ then for any $$\epsilon>0$$, i) If $$s\leq (1-\epsilon)s_0(d)$$, we have $P(\rho\text{ is separable})\leq 2\exp(-c(\epsilon)d^3),$ ii) If $$s\geq(1+\epsilon)s_0(d)$$, we have $P(\rho\text{ is entangled})\leq 2\exp(-c(\epsilon)s)."$

##### MSC:
 81P40 Quantum coherence, entanglement, quantum correlations 81P45 Quantum information, communication, networks (quantum-theoretic aspects)
##### Keywords:
quantum states; separability; entanglement
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