Quantum state tomography for generic pure states

Abstract

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 H_A ⊗ H_B ⊗ H_C, where dim(H_A) = 2 and dim(H_B) = dim(H_C), can be uniquely determined by its reduced states on subsystems H_A ⊗ H_B ⊗ H_C. 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 = d^N 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.