Tensor network decompositions of absolutely maximally entangled states

Consider a Hilbert space of N spins with local dimension D. Absolutely maximally entangled states (AME) are special vectors in this Hilbert space, which have maximal entanglement for all possible bi-partitions of the spins. There are many known constructions and examples (and sometimes even non-existence theorems) for AME with varying values of N and D. In this talk we consider the question, whether AME can be constructed from tensor networks with only a few local tensors with 4 legs. The question is equivalent to asking: how many two-site gates are needed to create an AME from a product state? In other words: how many local steps are needed to create maximal entanglement? We show that in certain cases an AME can be constructed with unexpectedly few local tensors. Concrete examples are for N=6 and N=8, and for various values of D>=5.