Analytic Expressions for Geometric Measure of Three Qubit States

A new method is developed to derive an algebraic equations for the geometric measure of entanglement of three qubit pure states. The equations are derived explicitly and solved in cases of most interest. These equations allow oneself to derive the analytic expressions of the geometric entanglement measure in the wide range of the three qubit systems, including the general class of W-states and states which are symmetric under permutation of two qubits. The nearest separable states are not necessarily unique and highly entangled states are surrounded by the one-parametric set of equally distant separable states. A possibility for the physical applications of the various three qubit states to quantum teleportation and superdense coding is suggested from the aspect of the entanglement.Comment: 6 pages, no figure, PRA versio

Mixed-State Entanglement and Quantum Teleportation through Noisy Channels

The quantum teleportation with noisy EPR state is discussed. Using an optimal decomposition technique, we compute the concurrence, entanglement of formation and Groverian measure for various noisy EPR resources. It is shown analytically that all entanglement measures reduce to zero when $\bar{F} \leq 2/3$, where $\bar{F}$ is an average fidelity between Alice and Bob. This fact indicates that the entanglement is a genuine physical resource for the teleportation process. This fact gives valuable clues on the optimal decomposition for higher-qubit mixed states. As an example, the optimal decompositions for the three-qubit mixed states are discussed by adopting a teleportation with W-stateComment: 18 pages, 4 figure

Completely mixed state is a critical point for three-qubit entanglement

Pure three-qubit states have five algebraically independent and one algebraically dependent polynomial invariants under local unitary transformations and an arbitrary entanglement measure is a function of these six invariants. It is shown that if the reduced density operator of a some qubit is a multiple of the unit operator, than the geometric entanglement measure of the pure three-qubit state is absolutely independent of the polynomial invariants and is a constant for such tripartite states. Hence a one-particle completely mixed state is a critical point for the geometric measure of entanglement.Comment: two references are added, reshaped, few points are clarifie