Only n-Qubit Greenberger-Horne-Zeilinger States are Undetermined by their Reduced Density Matrices

TL;DR

This paper proves that only generalized n-qubit GHZ states are not uniquely identified by their reduced density matrices, highlighting their unique information content at the n-party level.

Contribution

It establishes a precise characterization of states undetermined by their reduced density matrices, linking this property to local unitary stabilizer subgroups.

Findings

Only n-qubit GHZ states are undetermined by their (n-1)-qubit reduced density matrices.

A connection between stabilizer subgroups and state determination is identified.

Generalized GHZ states uniquely contain n-party information among pure states.

Abstract

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.