Deriving the Qubit from Entropy Principles

We provide an axiomatization of the simplest quantum system, namely the qubit, based on entropic principles. Specifically, we show: The qubit can be derived from the set of maximumentropy probabilities that satisfy an entropic version of the Heisenberg uncertainty principle. Our formulation is in phase space (following Wigner [41]) and makes use of Rényi [32] entropy (which includes Shannon [33] entropy as a special case) to measure the uncertainty of, or information contained in, probability distributions on phase space. We posit three axioms. The Information Reality Principle says that the entropy of a physical system, as a measure of the amount or quantity of information it contains, must be a real number. The Maximum Entropy Principle, well-established in information theory, says that the phase-space probabilities should be chosen to be entropy maximizing. The Minimum Entropy Principle is an entropic version of the Heisenberg uncertainty principle and is a deliberately chosen physical axiom. Our approach is thus a hybrid of information-theoretic (“entropic”) and physical (“uncertainty principle”) axioms.