Modes, states and superselection rules in quantum optics and quantum information (2501.03943v4)

Abstract: A convenient way to represent quantum optical states is through the quadrature basis of single-modes of the field. This framework provides intuitive definitions for quasi-classical states, their phase-space representations, and enables the definition of a universal gate set. In this widely adopted representation of quantum optics, most pure states consist of coherent superpositions of photon-number states. However, this approach neglects the particle-number superselection rule - which prohibits coherence between states of differing photon numbers - and implicitly assumes a phase reference. We adopt a representation of quantum optical states that respects the superselection rule and revisit key tools and results in quantum optics and information encoding within quantum optics. This approach preserves the intuitive aspects of the traditional quadrature representation while unifying insights from quantum optics with those from symmetric spin-like and angular momentum systems. More than just an alternative representation, we show that a superselection rule-compliant framework provides a unified formalism for all bosonic encodings, from single-photon to continuous-variable encodings. This perspective allows for a precise characterization of the roles of Gaussian and non-Gaussian resources, as well as the interplay between modes and states in quantum universality and potential computational advantage.