Only $n$-Qubit Greenberger-Horne-Zeilinger States Are Undetermined by Their Reduced Density Matrices

The generalized $n$-qubit Greenberger-Horne-Zeilinger (GHZ) states and their local unitary equivalents are the only states of $n$ qubits that are not uniquely determined among pure states by their reduced density matrices of $n-1$ qubits. Thus, among pure states, the generalized GHZ states are the only ones containing information at the $n$-party level. We point out a connection between local unitary stabilizer subgroups and the property of being determined by reduced density matrices.