Quantum information with general quantum variables: a formalism encompassing qubits, qudits, and quantum continuous variables

Abstract

This note presents a simple and unified formulation of the most fundamental structures used in quantum information with qubits, arbitrary dimension qudits, and quantum continuous variables. This \emph{general quantum variables} construction provides a succinct language for formulating many results in quantum computation and information so that they are applicable in all dimensions. The structures included within this formalism include: a generalization to arbitrary dimension of the three Pauli operators, and the associated mutually unbiased bases; the Pauli and Clifford groups; many important quantum gates; standard sets of generators for the Clifford group; and simple universal gate sets. This formalism provides a convenient, intuitive and extensible language for easily generalizing results that were originally derived for a single type of system (often qubits or quantum continuous variables), and that rely on only those structures listed above, to apply in all dimensions.