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 quantumcomputing.SE

• The article I'm mainly trying to understand is this arXiv:2001.07050, while the article with the quote above (which is helping me understand some background material) is arXiv:1905.10505v2 . So based on your answer the term genuine is just a classification of multipartite entanglement? Apr 12 '20 at 20:18
• It could refer to the observation that, with more than 2 particles, you can have only some of them entangled, i.e $1$ and $2$ entangled but $3$ separable from $1$ and $2$. There is no separable subsystem in a GHZ state. Apr 12 '20 at 20:43
• I'm not sure there is a clean definition; in the context of multipartite entanglement it typically means - I believe - multipartite entanglement which cannot understood as being made from "simpler" (less multipartite) entanglement (such as a tripartite state where only A&B are entangled). -- N.B.: This also explains why it only shows up for n>2. -- But my feeling is that "genuine" is at the same formal level as "generic" in quite a few cases. Apr 12 '20 at 21:19
• So follow up question that will help illustrate a lot to me: Suppose I have a composite Hilbert space $H = H_A \otimes H_B \otimes H_C$ and suppose all the subsystems A,B,C's Hilbert spaces are of dim = 2. Then let $|\psi\rangle$ be a state in $H_A$ and $|\Phi\rangle$ be any Bell/EPR state in $H_B \otimes H_C$ then define $|\phi\rangle = |\psi\rangle \otimes |\Phi\rangle$, so is $|\phi\rangle$ an entangled tripartite state but not a GENUINE tripartite entangled state because it's biseparable? Apr 12 '20 at 23:26
• @LostInEuclids5thPostulate Pretty much, yes. Apr 13 '20 at 14:57