Quantum State Tomography for Generic Pure States

Abstract

We show that a generic $N$-qudit pure quantum state is uniquely determined by only $2$ of its $\lceil \frac{N+1}{2}\rceil$-particle reduced density matrices. Therefore we give a method to uniquely determine a generic $N$-qudit pure state of dimension $D=d^N$ with $O(D)$ local measurements, which is an improvement comparing with the previous known approach using $O(D{\log }^{2}D)$ or $O(D\log D)$ measurements.