On the connection between weak measurement in quantum physics and analytic phase-retrieval in classical wave optics

The physical interpretation of weak measurements has been the subject of much debate. It is known that anomalous phenomena and results that appear in weak measurements are essentially related to the phase of the quantum system being measured. Consideration of the phase is important to clarify its physical interpretation. In classical wave optics, there has long been studies on methods of measuring or retrieving the phase of a wave function. We here present that one of those methods, the analytic phase retrieval based on the properties of entire functions, has a close connection with weak measurements in quantum physics. We explain such a connection for two emblematic optical weak-measurements that have the same mathematical formalism as quantum systems: one is a system for weak measurements of polarized light displacement in a birefringent crystal, and the other is a system for the direct measurement of a wave function by weakly coupling it to a pointer. In those two systems, we show that the pre- and post-selection of polarized light provides a filtering effect similar to that utilized in the analytic phase retrieval.