Geometrical interpretation of the density matrix: Mixed and entangled states

Abstract

The correspondence between the density matrices ρ N × N and the points in {ie564-01} is clarified. The particular cases of N = 2 and N = 4 are considered in detail. The geometrical representation of pure and mixed states is given. A mixture characteristic is introduced and its relation to Tr ρ 2 is obtained. The measure of distance between two quantum states is established in a natural way, in view of such an approach. Attention is paid to the geometrical description of tomograms, the distance between them, and its relation to the geometrical distance between the density matrices. The entanglement of a system of two qubits is investigated in the geometrical picture, which makes the boundary between separable and entangled states take the form of algebraic inequalities. An easily computable measure of entanglement is suggested and its physical basis is discussed, resulting in a generalization of the notion of fidelity to mixed states. The measure of entanglement obtained for the Werner state is compared with the entanglement of creation, the relative entropy of entanglement, and other characteristics involved in the entanglement problem. For the pure states, a comparison with the approach employing the von Neumann entropy is discussed as well.