Power of symmetric extensions for entanglement detection

In this paper, we present progress on the study of the symmetric extension criterion for separability. First, we show that a perturbation of order O(1/N) is sufficient and, in general, necessary to destroy the entanglement of any state admitting an N Bose-symmetric extension. On the other hand, the minimum amount of local noise necessary to induce separability on states arising from N Bose-symmetric extensions with positive partial transpose (PPT) decreases at least as fast as O(1/N²). From these results, we derive upper bounds on the time and space complexity of the weak membership problem of separability when attacked via algorithms that search for PPT-symmetric extensions. Finally, we show how to estimate the error we incur when we approximate the set of separable states by the set of (PPT) N-extendable quantum states in order to compute the maximum average fidelity in pure state estimation problems, the maximal output purity of quantum channels, and the geometric measure of entanglement.