Hierarchical graph states of k-uniform and absolutely maximally entangled states

It is important to characterize the entanglement properties of multipartite quantum states such as graph states, k-uniform, and Absolutely-Maximally-Entangled (AME) states. By merging these concepts we provide a remarkable relationship between geometry of graph states and entanglement of the corresponding states.

The k-uniform states are a particular type of highly entangled states, all of their reductions to k parties are maximally-mixed. A special case of this set are AME states that are maximally entangled along any splitting of parties into two groups. Moreover, the graph states are another set of well-entangled quantum states defined based on a graph, are also a class of stabilizer states.

We present a hierarchical procedure that leads us to study the geometric structure of k-uniform and AME states in the graph states representation. First, we introduce the most generalized method of constructing k-uniform and AME states from classical-error-correcting codes. Next, we introduce hierarchical graph states that construct the highly entangled states by iterating a certain geometric structure in graph state representation. We also show at each level of iteration the states that are all k-uniform cannot be converted into each other by SLOCC (stochastic local operations and classical communication).