Absolutely maximally entangled states of seven qubits do not exist

Felix Huber, Otfried Gühne, Jens Siewert

Pure multiparticle quantum states are called absolutely maximally entangled
if all reduced states obtained by tracing out at least half of the particles
are maximally mixed. We provide a method to characterize these states for a
general multiparticle system. With that, we prove that a seven-qubit state
whose three-body marginals are all maximally mixed, or equivalently, a pure
$((7,1,4))_2$ quantum error correcting code, does not exist. Furthermore, we
obtain an upper limit on the possible number of maximally mixed three-body
marginals and identify the state saturating the bound. This solves the
seven-particle problem as the last open case concerning maximally entangled
states of qubits.