The PPT square conjecture holds generically for some classes of independent states

Let be a random pure state on , where ψ is a random unit vector uniformly distributed on the sphere in . Let be a random induced state whose distribution is , and let be a random induced state following the same distribution independent of . Let ρ be a random state induced by the entanglement swapping of and . We show that the empirical spectrum of converges almost surely to the Marcenko–Pastur law with parameter c 2 as and . As an application, we prove that the state ρ is separable generically if are PPT entangled.