General stabilizer approach for constructing highly entangled graph states

Highly entangled multipartite states such as $k$-uniform ($k$-UNI) and absolutely maximally entangled (AME) states serve as critical resources in quantum networking and other quantum information applications. However, there does not yet exist a complete classification of such states, and much remains unknown about their entanglement structure. Here we substantially broaden the class of known $k$-UNI and AME states by introducing a method for explicitly constructing such states that combines classical error correcting codes and qudit graph states. This method in fact constitutes a general recipe for obtaining multipartitite entangled states from classical codes. Furthermore, we show that at least for a large subset of the class of $k$-UNI states that we present, the states are inequivalent under stochastic local operations and classical communication. This subset is defined by an iterative procedure for constructing a hierarchy of $k$-UNI graph states.