Absolutely maximally entangled states of seven qubits do not exist

TL;DR

This paper proves that absolutely maximally entangled states of seven qubits do not exist, resolving a long-standing open problem in quantum information theory by establishing the non-existence of such states.

Contribution

It introduces a general method to characterize absolutely maximally entangled states and proves the non-existence of seven-qubit states with maximally mixed three-body marginals.

Findings

Seven-qubit absolutely maximally entangled states do not exist.

Established an upper limit on maximally mixed three-body marginals.

Solved the last open case in maximally entangled states of qubits.

Abstract

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.