On the Geometry of Entanglement

Michel Boyer, Rotem Liss, Tal Mor

Published 2016-08-02Version 1

Entanglement is an important concept in quantum information and computing. Here we provide a simple and elegant geometrical analysis of all rank-2 quantum mixed states. The analysis is complete for all the bipartite states, and is partial for all the multipartite states. We characterize each rank-2 mixed state by defining the Bloch sphere spanned by the eigenstates of the density matrix describing the mixed state, and by the state's location within its Bloch sphere. We provide various interesting examples of entanglement and separability of mixed states belonging to exactly five classes of Bloch spheres, and we prove that there are no other classes. The tools developed in our work can be used for analysing the entanglement of all the quantum states that are "neighbours" of a rank-2 quantum state (namely, of all the states in its Bloch sphere).