Unified framework for bosonic quantum information encoding, resources, and universality from superselection rules

Since the early days of quantum information science, diverse strategies have been proposed to encode qubits and implement quantum gates across various physical platforms. Modes and bosonic states offer a flexible framework for quantum information processing, from single-photon optics to superconducting circuits, combining massless and massive particles. Universal gate sets have been established for specific encodings—such as those based on Fock states or the field’s quadratures. However, a general framework clarifying how different bosonic resources—such as mode structure, particle-number statistics, Gaussian and non-Gaussian states and operations—combine to enable universality and potential quantum advantage irrespectively of the encoding remains lacking. A main difficulty lies in the coexistence of distinct paradigms that appear to combine loosely in bosonic quantum information: discrete-variable schemes—where information is encoded in qubits or effective finite-dimensional systems—and continuous-variable schemes—which rely on infinite-dimensional Hilbert spaces associated with the field’s quadratures. Despite the success of each approach, they remain seemingly incompatible in many aspects—in particular in terms of physical resource analysis for universality—due to their underlying mathematical differences. In this work, we apply a superselection-rule-compliant framework for bosonic quantum information that unifies these paradigms. This representation explicitly incorporates the phase reference and enforces total particle number conservation, allowing for a consistent treatment of bosonic quantum information protocols and resources across both discrete and continuous regimes. We derive necessary and sufficient conditions expressed in terms of Gaussian and non-Gaussian resources for universality that apply to any encoding, without resorting to specific mappings. Our results show that continuous variable encodings are not fundamentally distinct from discrete ones, but rather emerge as subspaces within a general bosonic structure. Importantly, in such a widely studied framework, non-Gaussian properties of bosonic states and operations can be washed out, providing evidence that the continuous variables regime is not the most suitable to fully assess the quantum properties of bosons.