Lewenstein-Sanpera decomposition of a generic 2x2 density matrix by using Wootters's basis

Abstract

The Lewenstein-Sanpera decomposition for a generic two-qubit density matrix is 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 $\sqrt{\sqrt{\rho}\tilde{\rho}\sqrt{\rho}}$ >. It is shown that a given density matrix $\rho$ with corresponding set of positive numbers $\lambda_i$ and Wootters's basis can transforms under $SO(4,c)$ into a generic $2\times2$ 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, $\rho$ can be represented as coset space $SO(4,c)/SO(4,r)$ together with positive numbers $\lambda_i$. By giving an explicit parameterization we characterize a generic orbit of group of local unitary transformations.