On the Local Distinguishability of Three Finite Photon Ensembles with Identical Compressed Density Operators (2025)
Quantum state discrimination, which seeks to correctly determine the state of a quantum system among a set of possible states through a finite number of measurements, has become fundamental for the increasing development of quantum technologies due to its underlying relation to various quantum information processing and communication tasks. In this work, we thoroughly explore the problem of distinguishing between two and three finite photon ensembles with identical compressed density operators, with a main focus on local solutions, i.e., with no entangling operations among ensemble photons. Using semidefinite programming, we provide optimal analytical and numerical discrimination strategies in the very low photon-number regime. Then, multiple approaches to an interferometry-based solution for the three-ensemble problem are analyzed based on existing experimental literature in single-photon polarization state discrimination. However, none of these approaches outperform a trivial extension to a well-known local solution of the two-ensemble problem given by d'Espagnat. An in-depth discussion on the local distinguishability of the three ensembles is given throughout the text. The results from these analyses suggest that more successful discrimination protocols require performing entangling operations among ensemble photons in order to exploit the inner correlations within each ensemble.