Composite quantum states can be classified by how they behave under local
unitary transformations. Each quantum state has a stabilizer subgroup and a
corresponding Lie algebra, the structure of which is a local unitary invariant.
In this paper, we study the structure of the stabilizer subalgebra for n-qubit
pure states, and find its maximum dimension to be n-1 for nonproduct states of
three qubits and higher. The n-qubit Greenberger-Horne-Zeilinger state has a
stabilizer subalgebra that achieves the maximum possible dimension for pure
nonproduct states. The converse, however, is not true: we show examples of pure
4-qubit states that achieve the maximum nonproduct stabilizer dimension, but
have stabilizer subalgebra structures different from that of the n-qubit GHZ
state.Comment: 6 page