Maximum stabilizer dimension for nonproduct states
Scott N. Walck, David W. Lyons · Physical Review A · 2007
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 $n\ensuremath{-}1$ 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.