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Using techniques from symplectic geometry, we prove that a pure state ofthree qubits is up to local unitaries uniquely determined by its one-particlereduced density matrices exactly when their ordered spectra belong to theboundary of the, so called, Kirwan polytope. Otherwise, the states with givenreduced density matrices are parameterized, up to local unitary equivalence, bytwo real variables. Given inevitable experimental imprecisions, this means thatalready for three qubits a pure quantum state can never be reconstructed fromsingle-particle tomography. We moreover show that knowledge of the reduceddensity matrices is always sufficient if one is given the additional promisethat 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.

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