# How to calculate the number of ebits in a graph state?

Given an arbitrary graph state $|G\rangle$ represented by the graph $G$, can one use the graphical structure to calculate the number of ebits (entanglement bits) present in the state?

If so, how?

• What, precisely, do you mean by the number of ebits present in the state? For example, do you mean the minimum number of entangled pairs required to make the state (use local unitaries to convert to the LU-equivalent graph with the minimum number of edges, and it's the number of edges), or something about how many can be extracted under some constraints (single copy, asymptotic limit, local operations only, operations local under bipartitions of the graph,...)? Sep 12 '18 at 8:53
• The former. That was my intuition, but I couldn't find a reference to prove so. Do you know of one? Sep 12 '18 at 11:26
• Now that I think about it, it's not so simple! Sep 12 '18 at 13:15
• Could you explain your thoughts on this? I think they would be constructive. Sep 12 '18 at 14:53

The number of Bell pairs required to construct a given graph state can easily be given an upper bound: $|V|-1$, where $V$ is the set of vertices. You do this simply by preparing the entire state at one site, and teleporting all the other qubits to the relevant party.