Separability And Measure Of Entanglement For Quantum States

Quantum entanglement has played a very important role in quantum information. However the physical characteristics and mathematical structures of entangled states have not been well understood. We investigate separability and measure of entanglement for quantum states. On the one hand, we devote to find methods to characterize the separable state 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 entangled states has been found. Firstly we investigate separability of mixed states via positive operator. The density matrix of a pure state is given by ρ =1/2(2/NI_N + rλ), where I_n is identity matrix, λ = (λ₁,..., λ_N²-1) and λ₁,..., λ_N²-1) are generators of SU(N). It is well known that when r is a real 3—dimensional vector, i.e. a Bloch vector, a state p is pure if and only if |r| = 1. But the relation between ρ and |r| is not clear for the case that r is a real (N² - 1)—dimensional vector. We present an analytical expression of the relation. Based on this conclusion we show that some quantity in relation to Hermitian matrix is positive if a quantum state in a bipartite system is separable. Furthermore we investigate separability of multipartite quantum mixed states. If a quantum state in a multipartite system is separable, we show that some quantity in relation to Hermitian matrix which is different from the case of bipartite systems is positive. Moreover we investigate separability of mixed states in bipartite and multipartite quantum systems via trace norm. We first consider the separability of mixed states in R₁× R₂ quantum systems. Let Γ_ρ denote the matrix with entries given by R_j1j2¹² =(R₁R₂)/4Tr(ρA₁A₂λ_j1¹Ã—λ_j2²) , where λ_j1¹ and λ_j2² are generators of SU(R₁) and SU(R₂), respectively. We show that trace norm of Γ_ρ is less than or equal to a constant if a quantum state is separable. Then we generalize the resultsto multipartite quantum systems.On the other hand, we consider measure of entanglement. A measure to quantify the degree of entanglement for two qubits in a pure state is presented in Phys. Rev. A 65, 044303 (2002) by J.L. Chen et al. We generalize their results to 2 × N quantum systems. An explicit lower bound of the entanglement of formation for quantum mixed states of 2 × N quantum systems is obtained. Moreover for bipartite and multipartite quantum systems a generalized "measure" is respectively presented to judge if a pure state is separable or maximally entangled. For bipartite pure state, a good measure of entanglement is the von Neumann entropy. In recent years, a number of measure of entanglements to quantify entanglement has been proposed. We mainly discuss the entanglement of formation. Although the entanglement of formation is defined for arbitrary dimensional bipartite systems, so far no explicit analytic formulae for entanglement of formation have been found for systems of dimension larger than 2, except for some special symmetric states. We study the entanglement of formation for higher dimensional quantum mixed states and discuss three cases. Let A denote the matrix with entries given by a_ij in the form of We first consider the case that AA⁺ has only two nonzero eigenvalues with the same algebraic multiplicity. We present the conditions allowing to derive an explicit lower bound of the entanglement of formation for such kind of arbitrary dimensional mixed states and calculate the lower bound. Then we consider the others cases. One case is that AA⁺ has only two non-zero eigenvalues with different algebraic multiplicity, the other case is that AA⁺ has many non-zero eigenvalues with the same algebraic multiplicity.