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We examine the problem of whether a multipartite pure quantum state can be uniquely determined by its reduced density matrices. We show that a generic pure state in three party Hilbert space HA ⊗ HB ⊗ HC, where dim(HA) = 2 and dim(HB) = dim(HC), can be uniquely determined by its reduced states on subsystems HA ⊗ HB ⊗ HC. Then, we generalize the conclusion to the case that dim(H1) > 2. As a corollary, we show that a generic N-qudit pure quantum state is uniquely determined by only two of its \(\left\lceil {\frac{{N + 1}}{2}} \right\rceil \)-particle reduced density matrices. Furthermore, our results indicate a method to uniquely determine a generic N-qudit pure state of dimension D = dN with only O(D) local measurements, which is an improvement compared to the previous known approach that uses O(Dlog2D) or O(Dlog D) local measurements.

Quantum state tomography for generic pure states