Characterization of High-Dimensional Unitary Transformations via Quantum Interferometry

We present a quantum interferometric method to measure high-dimensional unitary transformations which are applicable to photonic single-qudit states. The method relies on interference by path identity of undetected photons. It utilizes a photon pair entangled in orbital angular momentum (OAM). The photon pair consists of signal and idler photons. The unknown unitary transformation is applied to the idler photon and reconstructed from single-photon interference patterns created by the signal photon, requiring neither coincidence measurements nor postselection. Unlike conventional techniques, our method does not involve direct measurement of the qudit undergoing the transformation. This provides a significant advantage in scenarios where the signal wavelength is easily detectable, but adequate detectors for the idler-wavelength are unavailable.