Quantifying Entanglement of Multipartite Quantum States

In this study, it is investigated that how to quantify and characterize the quantum states of n-particles from the point of (k+1)-partite entanglement (1 <= k <= n-1), which plays an instrumental role in quantum nonlocality and quantum metrology. Two families of entanglement measures termed q-(k+1)-PE concurrence (q>1) and alpha-(k+1)-PE concurrence (0 <=alpha<1) are put forward, respectively. As far as the pure state is concerned, they are defined based on the minimum in entanglement. Meanwhile, rigorous proofs showing that both types of quantifications fulfill all the requirements of an entanglement measure are provided. In addition, two alternative kinds of entanglement measures are also proposed, named q-(k+1)-GPE concurrence (q > 1) and alpha-(k+1)-GPE concurrence (0 <= alpha < 1), respectively, where the quantifications of any pure state are given by taking the geometric mean of entanglement under all partitions satisfying preconditions. Besides, the lower bounds of these measures are presented by means of the entanglement of permutationally invariant (PI) part of quantum states, and the interrelations of these measures are elucidated. Moreover, these measures are compared and explained the similarities and differences among them. Furthermore, for computational convenience, enhanced versions of the quantifications mentioned above that can be utilized to distinguish whether a multipartite state is genuinely strong k-producible are considered.