Absolutely Maximally Entangled Qudit Graph States

Absolutely maximally entangled (AME) states are multipartite entangled states that are maximally entangled for any possible bipartition. In this paper, we study the description of AME states within the graph state formalism. The graphical representation provides an intuitive framework to visualize the entanglement in graph states, which makes them a natural candidate to describe AME states. We show two different methods of determining bipartite entanglement in graph states and use them to define various AME graph states. We further show that AME graph states exist for all number of parties, and that any AME graph states shared between an even number of parties can be used to describe quantum secret sharing schemes with a threshold or ramp access structure directly within the graph states formalism.