# What is the definition of genuine entanglement?

I can't seem to find a definition for genuine entanglement but it is being used in an article I'm reading and the references inside but I'm unable to find the precise definition.

One thing I was able to get to the bottom of is "So far, genuine multipartite entangled (GME) states in the form of Greenberger-HorneZeilinger (GHZ) states..", but I do not know what about the GHZ states makes them "genuinely entangled".

Also the way it comes up is "genuine multipartite entanglement", which begs the questions (after what is genuine entanglement) why does it only come up in multipartite schemes ($$n>2$$)?

Cross-posted on physics.SE

• I'm voting to close this question as off-topic because it was also asked on Physics SE and it is discussed there. Here is a link: physics.stackexchange.com/questions/543411/… Commented Apr 12, 2020 at 22:39
• @MartinVesely see this discussion on meta about cross-posting (you might want to argue about whether we should not allow it?)
– glS
Commented Apr 13, 2020 at 14:59
• I think it should be allowed just because of the difference in the overlap of people. On Physics stack more often people will give purely physically motivated answers/references while here people will more likely give a quantum info motivated answer/reference. Commented Apr 13, 2020 at 18:16
• I agree, I retracted my close vote. Commented Apr 13, 2020 at 18:18

## 1 Answer

Following Gühne and Toth (2008) (pag. 18), a pure state is genuinely multipartite entangled if it cannot be written as tensor product of two states in any bipartition. Equivalently, a state is multipartite entangled if all reduced states are mixed (i.e. computing the partial trace with respect to any subsystem gives a mixed state).

There is only one way to partition a bipartite space, thus the notion of "genuine" bipartite entanglement is equivalent to that of "normal" entanglement.