Negativity as a distance from a separable state

The computable measure of the mixed-state entanglement, the negativity, is shown to admit a clear geometrical interpretation, when applied to Schmidt-correlated (SC) states: the negativity of a SC state equals a distance of the state from a pertinent separable state. As a consequence, the Peres-Horodecki criterion of separability is both necessary and sufficient for SC states. Another remarkable consequence is that the negativity of a SC can be estimated “at a glance” on the density matrix. These results are generalized to mixtures of SC states, which emerge in bipartite evolutions with additive integrals of motion.

Figure 1

The negativity of a SC state $\widehat{\rho }$ equals half the sum of the absolute values of the off-diagonal elements of the density matrix, which is a distance of the state state from the separable state ${\widehat{\rho }}^{\prime }$.