Lewenstein-Sanpera decomposition of a generic 2 circle times 2 density matrix by using Wootters's basis
An analytical expression for optimal Lewenstein-Sanpera (L-S) decomposition of a generic two qubit density matrix is given. By evaluating the L-S decomposition of Bell decomposable states, the optimal decomposition for arbitrary full rank state of two qubit system is obtained via local quantum operations and classical communications (LQCC). In Bell decomposable case the separable state optimizing L-S decomposition, minimize the von Neumann relative entropy as a measure of entanglement. The L-S decomposition for a generic two-qubit density matrix is only obtained by using Wootters's basis. It is shown that the average concurrence of the decomposition is equal to the concurrence of the state. It is also shown that all the entanglement content of the state is concentrated in the Wootters's state \x(1)> associated with the largest eigenvalue lambda(1) of the Hermitian matrix rootrootrho(&rho;) over tilde rootrho. It is shown that a given density matrix p with corresponding set of positive numbers lambda(i) and Wootters's basis can transforms under SO(4,c) into a generic 2 x 2 matrix with the same set of positive numbers but with new Wootters's basis, where the local unitary transformations correspond to SO(4,r) transformations, hence, p can be represented as coset space SO(4, c)/SO(4, r) together with positive numbers lambda(i).
lewenstein-sanpera decomposition bell decomposable states lqcc concurrence quantum entanglement wootters' basis