Measurement of high-dimensional unitary operations without detecting the qudit they transform

Measurement of high-dimensional unitary transformations is essential for numerous tasks in quantum information science. Standard methods of measuring such transformations require measuring the qudits to which the transformations are applied. For photonic systems, this often poses a major challenge due to the lack of adequate single-photon detectors across all spectral ranges, especially in the mid- and far-infrared regions. We propose an interferometric method to measure an arbitrary high-dimensional unitary transformation which, in contrast with existing methods, does not require measuring the qudit to which the transformation is applied. Our method relies on quantum interferometry, namely interference by path identity of undetected photons. Here, an interferometric measurement is performed on a second qudit that is correlated with the qudit transformed by the unitary operation; no coincidence measurement is required. Working with orbital angular momentum (OAM) states of light, we demonstrate that the unitary transformation can be fully reconstructed from single-photon interference patterns without any postselection.