Only $n$-qubit Greenberger-Horne-Zeilinger states contain $n$-partite information

The generalized $n$-qubit Greenberger-Horne-Zeilinger (GHZ) states and their local unitary equivalents are the only pure states of $n$ qubits that are not uniquely determined (among arbitrary states, pure or mixed) by their reduced density matrices of $n-1$ qubits. Thus, the generalized GHZ states are the only ones containing information at the $n$-party level.