Variable Beam-Splitter Reflectivity Estimation for Interferometry

Abstract

Quantum parameter estimation is the scientific study that involves the use of quantum measurements for estimating an unknown parameter. Quantum-enhanced adaptive phase estimation is a well-studied example where the goal is to estimate the unknown phase such that the phase imprecision scales better than the standard quantum limit. Geometrically speaking, phase estimation is a U(1) estimation problem where the objective is to make a highly precise estimate of the rotation of the initial state about the ordinate in the state space. This measurement technique has been shown to give better precision than the same measurement done classically. Whereas other studies focus on phase estimation, we forge a new path that involves the variable beam-splitter reflectivity estimation. Our work proposes using quantum resources to solve the estimation problem. Here the phase is kept constant and the beam-splitter reflectivities are varied. We aim to achieve quantum-enhanced precision in the case of variable beam-splitter reflectivity estimation utilizing an evolutionary algorithm. In this thesis, we show that variable beam-splitter reflectivity estimation is also a U(1) rotation problem involving a different U(1) subgroup. Geometrically speaking, the goal of the research work is to estimate the angle between the abscissa and the unknown axis in the equatorial plane the initial state is rotated about. We devise an optimization algorithm that designs a policy that estimates the unknown beam-splitter reflectivity whose imprecision scales better than the standard quantum limit with respect to the photon number. We employ the differential evolutionary algorithm, inspired by genetic evolution, for policy search. Differential evolution is an optimization algorithm that performs feasibility variant of the non-convex optimization and finds optimal policies for quantum-enhanced precision. In our work, we propose injecting an N-photon permutationally symmetric input state known as the sine state, where N varies from 4 to 100. We do not include photon loss and noise in our model. Our work sets the stage for two-parameter and multi-parameter quantum-enhanced estimation schemes.