We show how to detect entangled, bound entangled and separable bipartite quantum states of arbitrary dimension and mixedness using geometric entanglement witnesses. These witnesses are constructed using properties of the Hilbert–Schmidt geometry and can be shifted along parameterized lines. The involved conditions are simplified using Bloch decompositions of operators and states. As an example we determine the three different types of states for a family of two-qutrit states that is part of the 'magic simplex', i.e. the set of Bell-state mixtures of arbitrary dimension.
We construct a simplex for multipartite qubit states of even number
n of qubits, which has the same geometry concerning separability, mixedness, kind of entanglement, amount of entanglement, and nonlocality as the bipartite qubit states. We derive the entanglement of the class of states which can be described by only three real parameters with the help of a multipartite measure for all discrete systems. We prove that the bounds on this measure are optimal for the whole class of states and that it reveals that the states possess only
n-partite entanglement and not, e.g., bipartite entanglement. We then show that this
n-partite entanglement can be increased by stochastic local operations and classical communication to the purest maximal entangled states. However, pure
n-partite entanglement cannot be distilled; consequently all entangled states in the simplex are
n-partite bound entangled. We study also Bell inequalities and find the same geometry as for bipartite qubits. Moreover, we show how the (hidden) nonlocality for all
n-partite bound entangled states can be revealed.
Li-Qiao Entanglement Constraints Confirming the Entanglement Probability of
2713 for the Two-Qutrit Hiesmayr-Löffler Magic Simplex of Bell States
We compute the measure of entanglement for some classes of states belonging to the simplex of Bell-diagonal states of two qutrits introduced by Baumgartner et al. [Phys. Rev. A 74, 032327 (2006)]. We discuss also partial characterization of that simplex in terms of negativity and linear entropy. A numerical exploration of the entropy-entanglement plane indicates that, in the analogy to the two-qubit case [Munro et al., Phys. Rev. A 64, 030302(R) (2001)], two-qutrit Werner states do not maximize entanglement for a given entropy.