Retrieving the vast amount of information carried by a photon is an enduring challenge in quantum metrology science and quantum photonics research. The transverse spatial state of a photon is a convenient high-dimensional quantum system for study, as
it has a well-understood classical analogue as the transverse complex field profile of an optical beam. One severe drawback of all currently available quantum metrology techniques is the need for a time-consuming characterization process, which scales very unfavorably with the dimensionality of the quantum system. Here we demonstrate a technique that directly measures a million-dimensional photonic spatial state in a single setting. Through the arrangement of a weak measurement of momentum and parallel strong measurements of position, the complex values of the entire photon state vector become measurable directly. The dimension of our measured state is approximately four orders of magnitude larger than previously measured. Our work opens up a practical route for characterizing high-dimensional quantum systems in real time. Furthermore, our demonstration also serve as a high-speed, extremely-high-resolution unambiguous complex field measurement technique for diverse classical applications.
Orbital angular momentum (OAM) of light is an attractive degree of freedom for funda- mentals studies in quantum mechanics. In addition, the discrete unbounded state-space of OAM has been used to enhance classical and quantum communications. Unambigu
ous mea- surement of OAM is a key part of all such experiments. However, state-of-the-art methods for separating single photons carrying a large number of different OAM values are limited to a theoretical separation efficiency of about 77 percent. Here we demonstrate a method which uses a series of unitary optical transformations to enable the measurement of lights OAM with an experimental separation efficiency of more than 92 percent. Further, we demonstrate the separation of modes in the angular position basis, which is mutually unbiased with respect to the OAM basis. The high degree of certainty achieved by our method makes it particu- larly attractive for enhancing the information capacity of multi-level quantum cryptography systems.
