Direct Measurement of Quantum State Rotations

The of a quantum state poses a unique challenge for experimentalists. Recently, the technique of measurement was proposed for characterizing a quantum state in-situ through sequential weak and strong measurements. While this method has been used for measuring polarization states, its real potential lies in the of states with a large dimensionality. Here we show the practical direct of a high-dimensional state vector in the discrete basis of orbital-angular momentum. Through weak measurements of orbital-angular momentum and strong measurements of angular position, we measure the complex probability amplitudes of a pure state with a dimensionality, d=27. Further, we use our method to directly observe the relationship between rotations of a state vector and the relative phase between its orbital-angular-momentum components. Our technique has important applications in high-dimensional classical and quantum information systems, and can be extended to characterize other types of large quantum states.