A Criterion to identify maximally entangled nine-qubit state

We present a generalized criterion for maximally entangled nine-qubit states, whose minimum averaged subsystem purity should be equal to 1/14. In this note, we prove that the absolutely maximally entangled state for nine qubits does not exist. In other words, we pronounce that the pure (9, 1, 5)₂ quantum error-correcting code does not exist. Further, we construct a new genuine nine-qubit maximally entangled states of 110,15 and 1 balanced purities equal to 1/16, 1/8 and 1/4, respectively. We found that the marginal density matrices for subsystems of 1,2,3-qubits all completely mixed in these states; therefore, maximally entangled nine-qubit state is a 3-uniform state.