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. 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.

Figure 1

The graph of the Fano (or seven-point) plane on the left, which can be transformed by local complementation (corresponding to local Clifford gates) to the wheel graph displayed on the right. The Fano plane plays a role in classical error correction, describing both a balanced block design as well as an error correcting code [31]. The corresponding graph state saturates the bound of Observation 3. The states are locally equivalent to the graph state depicted in Figs. 1 in [30], to No. 44 in Table V from Ref. [19], and to the states of Eq. (11) in Ref. [9] and of Eq. (26) in Ref. [23].