When can a pair of entangled particle become separable? I understand that quantum particles are easily disturbed by the noise of the environment. I would like to know that, in what condition at a pair of entangled particle becomes separable state? Is there a way to find out?
 A: Entanglement is a quite fragile resource, so the short answer to your question is “with enough noise”.
More quantitatively, you can picture the state space of two particles as follows, with a convex “island” of separable (i.e. non-entangled states) surrounded by a “sea” of entangled states. For most noise models, the maximal amount of noise leads to a separable state (often the maximally separable states). When you progressively add noise to an entangled state, you progressively move along the line linking it to the noisy final state. At some point, you cross the boundary between entangled and separable states, and you have no entanglement left.

The amount of noise needed to transform an entangled pair into a a separable one is closely linked with the robustness on entanglement, introduced in 1998 by Guifré Vidal and Rolf Tarrach (arXiv:quant-ph/9806094, Phys. Rev. A 59 141 and its generalized version, introduced by Michael Steiner in 2003 (arXiv:quant-ph/0304009/Phys. Rev. A 63 054305). These robustnesses measure (in two different settings) the amount of effort needed to transform an entangled state into a separable state.
