Bipartite entanglement

The Hohenberg-Kohn theorem can be used to redefine entanglement measures in terms of new physical quantities expectation values of observables, ai, instead of external control parameters, li. Consider an arbitrary entanglement measure M for the ground state of Hamiltonian (85). For a bipartite entanglement, one can prove a central lemma, which very generally connects M and energy derivatives. 

At the current time, there is no simple way to carry out the calculations with all these entanglement measures. Their properties, such as additivity, convexity, and continuity, and relationships are still under active investigation. Even for the best-understood entanglement of formation of the mixed states in bipartite systems AB, once the dimension or A or B is three or above, we don t know how to express it simply, although we have the general definitions given previously. However, for the case where both subsystems A and B are spin-i particles, there exists a simple formula from which the entanglement of formation can be calculated [42]. 

As we mentioned before, when a biparticle quantum system AB is in a pure state, there is essentially a unique measure of the entanglement between the subsystems A and B given by the von Neumann entropy S = —Tr[p log2 PaI- This approach gives exactly the same formula as the one given in Eq. (26). This is not surprising since all entanglement measures should coincide on pure bipartite states and be equal to the von Neumann entropy of the reduced density matrix (uniqueness theorem). 

QUBIT ENTANGLEMENT FROM A BIPARTITE ATOMIC SYSTEM UNDER STRONG ATOM-VACUUM-FIELD COUPLING IN A CARBON NANOTUBE... 

---