A Hierarchy of Entanglement

Časlav Brukner, Marek Żukowski, Anton Zeilinger

Nathaniel Johnston, Benjamin Lovitz, Aravindan Vijayaraghavan

We introduce a hierarchy of linear systems for showing that a given subspace of pure quantum states is entangled (i.e., contains no product states). This hierarchy outperforms known methods already at the first level, and it is complete in the sense that every entangled subspace is shown to be so at some finite level of the hierarchy. It straightforwardly generalizes to the case of higher Schmidt rank, as well as the multipartite cases of completely and genuinely entangled subspaces. These hierarchies work extremely well in practice even in very large quantum systems, as they can be implemented via elementary linear algebra techniques rather than the semidefinite programming techniques that are required by previously known hierarchies.

Bob Coecke

Wei Niu, Jiong Cheng, Wen-Zhao Zhang +1 more

Abstract We propose a criterion for detecting quantum states containing non-zero entanglement with respect to all possible bipartitions of a tripartite system. Our criterion, together with two other criteria, forms a hierarchy of tripartite entanglement detection, revealing three distinct types: genuine tripartite entanglement, entanglement with respect to all bipartitions, and standard tripartite entanglement. Furthermore, we present a feasible scheme for generating and detecting this hierarchy of tripartite entanglement within a cavity magnomechanical system. Our research paves the way for generating and detecting tripartite entanglement hierarchies in continuous variable systems.

Willi-Hans Steeb, Yorick Hardy

Abstract We consider a hierarchy of Hamilton operators Ĥ N in finite dimensional Hilbert spaces $$ \mathbb{C}^{2^N } $$. We show that the eigenstates of Ĥ N are fully entangled for N even. We also calculate the unitary operator U N(t) = exp(—Ĥ N t/ħ) for the time evolution and show that unentangled states can be transformed into entangled states using this operator. We also investigate energy level crossing for this hierarchy of Hamilton operators.