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Absolutely maximally entangled (AME) states of $k$ qudits (also known asperfect tensors) are quantum states that have maximal entanglement for allpossible bipartitions of the sites/parties. We consider the problem of whethersuch states can be decomposed into a tensor network with a small number oftensors, such that all physical and all auxiliary spaces have the samedimension $D$. We find that certain AME states with $k=6$ can be decomposedinto a network with only three 4-leg tensors; we provide concrete solutions forlocal dimension $D=5$ and higher. Our result implies that certain AME stateswith six parties can be created with only three two-site unitaries from aproduct state of three Bell pairs, or equivalently, with six two-site unitariesacting 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-legtensors.

Tensor network decompositions for absolutely maximally entangled states