Perfect tensors describe highly entangled quantum states that have attracted
    particular attention in the fields of quantum information theory and quantum
    gravity. In loop quantum gravity, the natural question arises whether SU(2)
    invariant tensors, which are fundamental ingredients of the basis states of
    spacetime, can also be perfect. In this work, we present a number of general
    constraints for the layout of such invariant perfect tensors (IPTs) and further
    describe a systematic and constructive approach to check the existence of an
    IPT of given valence. We apply our algorithm to show that no qubit encoding of
    valence 4 and 6 can be described by an IPT and close a gap to prove a no-go
    theorem for invariant perfect qubit encodings.

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