Quantifying entanglement with witness operators

Abstract

We present a unifying approach to the quantification of entanglement based on entanglement witnesses, which includes several already established entanglement measures such as the negativity, the concurrence, and the robustness of entanglement. We then introduce an infinite family of new entanglement quantifiers, having as its limits the best separable approximation measure and the generalized robustness. Gaussian states, states with symmetry, states constrained to super-selection rules, and states composed of indistinguishable particles are studied under the view of the witnessed entanglement. We derive new bounds to the fidelity of teleportation ${d}_{\mathit{min}}$, for the distillable entanglement ${E}_{D}$ and for the entanglement of formation. A particular measure, the PPT-generalized robustness, stands out due to its easy calculability and provides sharper bounds to ${d}_{\mathit{min}}$ and ${E}_{D}$ than the negativity in most of the states. We illustrate our approach studying thermodynamical properties of entanglement in the Heisenberg XXX and dimerized models.