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
\[
P(\rho\text{ is entangled})\leq 2\exp(-c(\epsilon)s)."
\]
Reviewer: Carlos F. Lardizabal (Porto Alegre)
MSC:
| 81P40 | Quantum coherence, entanglement, quantum correlations |
| 81P45 | Quantum information, communication, networks (quantum-theoretic aspects) |
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