  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.
