Quantum State Of The Separability And Entanglement Of Formation Degrees

We investigate separability and entanglement measure for quantum states. On the one hand, we devote to find methods to characterize separable states and present some necessary conditions for separability of mixed states. For pure states the separability is quite well understood. However, for the case of mixed states, no single practical procedure that can be guaranteed to detect the entanglement of every states has been found.Using the generator of SU(N), any bipartite state can be represented by ρ_AB =be the Bloch sphere representation ofρ_AB which is in the subsystem S(H_M), where (r₁,r₂···r_M²-1) ∈ R^M2-1 be a vectorof dim M² - 1. Then we get.For a bipartite separable state ρ_AB, which is of M × N dimension, it can be written in the following form: .By defining thetransformation R of M_i, we get a new operator γ_R(ρ_AB). We proved that the modular of the Bloch vector in the subsystem S(H_M) of ρ_AB is no more than the modular of the original Bloch vector, that is (|r_i|)|�'≤(|r_i|)|�'.For M=3, we extract a special transformation R and by the condition γ_R(ρ_AB)≥0 we get PPT criterion.We discuss the question of entanglement measure. For a bipartite pure state, a good measure of entanglement is the von Neumann entropy. For mixed states there is no unique entanglement measure. In recent years, a number of entanglement measures to quantify entanglement has been proposed. We mainly discuss the entanglement of formation. Under the lower dimension condition, Wootters et al proved that there is a monotone relation between the entanglement of formation and concurrence. We investigate a generator of the special unitary group SU(N) using the projector operators P_jk. By using these generators, we get the generalized...