Essentially entangled component of multipartite mixed quantum states, its properties, and an efficient algorithm for its extraction

Using geometric means, we first consider a density matrix decomposition of a multipartite quantum system of a finite dimension into two density matrices: a separable one, also known as the best separable approximation, and an essentially entangled one, which contains no product state components. We show that this convex decomposition can be achieved in practice with the help of a linear programming algorithm that scales in the general case polynomially with the system dimension. We illustrate the algorithm implementation with an example of a composite system of dimension 12 that undergoes a loss of coherence due to classical noise and we trace the time evolution of its essentially entangled component. We suggest a “geometric” description of entanglement dynamics and demonstrate how it explains the well-known phenomena of sudden death and revival of multipartite entanglements. For a statistical weight loss of the essentially entangled component with time, its average entanglement content is not affected by the coherence loss.

Figure 1

A symbolic illustration of the geometric structure of density matrices and of the decomposition Eq. (2).

Figure 2

A symbolic description of the trajectory in the Liouville space of a mixed state undergoing loss of coherence due to interaction with the environment. Crossing of the polytope of the separable states results in sudden death (or birth) of entanglement. The inset lists the numerical values of the parameters of the model.

Figure 3

We solve the Lindblad equation for the example and we apply the algorithm at each time step: (a) purity of the assembly, (b) the statistical contribution of ${\widehat{\rho }}_{\mathrm{ent}}\left(t\right)$ to the density matrix, (c) the rank of ${\widehat{\rho }}_{\mathrm{ent}}\left(t\right)$, (d) the eigenvalue of the dominant eigenvector of ${\widehat{\rho }}_{\mathrm{ent}}\left(t\right)$, and (e–h) the oscillations of the real coefficients of the tanglemeter. In the time interval $\left[18.8\text{–}19.7\right]$, sudden death of entanglement takes place and then its revival.