Tensor network decompositions for absolutely maximally entangled states

Absolutely maximally entangled (AME) states of k qudits are quantum states that have maximal entanglement for all possible bipartitions of the sites/parties, and they can also be interpreted as perfect tensors. We consider the problem of whether such states can be decomposed into a tensor network with a small number of tensors, such that all physical and all auxiliary spaces have the same dimension D. We find that certain AME states with k = 6 can be decomposed into a network with only three 4-leg tensors; we provide concrete solutions for local dimension D = 5 and higher. Our result implies that certain AME states with six parties can be created with only three two-site unitaries from a product state of three Bell pairs, or equivalently, with six two-site unitaries acting on a product state on six qudits. We also consider the problem for k = 8, where we find similar tensor network decompositions with six 4-leg tensors.