Deriving the Qubit from Entropy Principles

Abstract

The Heisenberg uncertainty principle is one of the most famous features of quantum mechanics. However, the non-determinism implied by the Heisenberg uncertainty principle --- together with other prominent aspects of quantum mechanics such as superposition, entanglement, and nonlocality --- poses deep puzzles about the underlying physical reality, even while these same features are at the heart of exciting developments such as quantum cryptography, algorithms, and computing. These puzzles might be resolved if the mathematical structure of quantum mechanics were built up from physically interpretable axioms, but it is not. We propose three physically-based axioms which together characterize the simplest quantum system, namely the qubit. Our starting point is the class of all no-signaling theories. Each such theory can be regarded as a family of empirical models, and we proceed to associate entropies, i.e., measures of information, with these models. To do this, we move to phase space and impose the condition that entropies are real-valued. This requirement, which we call the Information Reality Principle, arises because in order to represent all no-signaling theories (including quantum mechanics itself) in phase space, it is necessary to allow negative probabilities (Wigner [1932]). Our second and third principles take two important features of quantum mechanics and turn them into deliberately chosen physical axioms. One axiom is an Uncertainty Principle, stated in terms of entropy. The other axiom is an Unbiasedness Principle, which requires that whenever there is complete certainty about the outcome of a measurement in one of three mutually orthogonal directions, there must be maximal uncertainty about the outcomes in each of the two other directions.Comment: 8 pages, 3 figure