A Pure Multiparticle Quantum Error Correcting Code

Absolutely maximally entangled states of seven qubits do not exist

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 and prove that a seven-qubit state whose three-body marginals are all maximally mixed, or equivalently, a pure quantum error correcting code, does not exist.Furthermore, we obtain an upper limit on the possible number of maximally mixed three-bodymarginals and identify the state saturating the bound.