Characterizing and Measuring Multipartite Entanglement

The quantification of multipartite entanglement is an open and very challenging problem. An exhaustive definition of bipartite entanglement exists and hinges upon the von Neumann entropy and the entanglement of formation, but the problem of defining multipartite entanglement is more difficult and no unique definition exists: different definitions tend indeed to focus on different aspects of the problem, capturing different features of entanglement, that do not necessarily agree with each other. Moreover, as the size of the system increases, the number of measures (i.e. real numbers) needed to quantify multipartite entanglement grows exponentially. This work is motivated by the idea that a good definition of multipartite entanglement should stem from some statistical information about the system. We shall therefore look at the distribution of the purity of a subsystem over all bipartitions of the total system. As a measure of multipartite entanglement we will take a whole function: the probability density of bipartite entanglement between any two parts of the total system. According to our definition multipartite entanglement is large when bipartite entanglement (i) is large and (ii) does not depend on the bipartition, namely when (i+ii) the probability density of bipartite entanglement is a narrow function centered at a large value. This definition will be tested on two class of states that are known to be characterized by a large entanglement. We emphasize that the idea that complicated phenomena cannot be “summarized” in a single (or a few) number(s) was already proposed in the context of complex systems and has been also considered in relation to quantum entanglement. We shall focus on a collection of n qubits and consider a partition in two subsystems A and B, made up of nA and nB qubits (nA + nB = n), respectively. For