When is a pure state of three qubits determined by its single-particle reduced density matrices?

TL;DR

This paper investigates when a pure three-qubit state is uniquely determined by its single-particle reduced density matrices, revealing geometric conditions and implications for quantum state tomography.

Contribution

It provides a geometric criterion using symplectic geometry to determine when three-qubit states are uniquely identified by their single-particle spectra, and discusses extensions to more qubits.

Findings

States with spectra on the boundary of the Kirwan polytope are uniquely determined.

States with spectra inside the polytope are parameterized by two variables.

Knowledge of reduced states suffices if the state is not SLOCC-equivalent to GHZ.

Abstract

Using techniques from symplectic geometry, we prove that a pure state of three qubits is up to local unitaries uniquely determined by its one-particle reduced density matrices exactly when their ordered spectra belong to the boundary of the, so called, Kirwan polytope. Otherwise, the states with given reduced density matrices are parameterized, up to local unitary equivalence, by two real variables. Given inevitable experimental imprecisions, this means that already for three qubits a pure quantum state can never be reconstructed from single-particle tomography. We moreover show that knowledge of the reduced density matrices is always sufficient if one is given the additional promise that the quantum state is not convertible to the Greenberger--Horne--Zeilinger (GHZ) state by stochastic local operations and classical communication (SLOCC), and discuss generalizations of our results to an arbitary number of qubits.