{
  "id": "1311.3365",
  "version": "v5",
  "published": "2013-11-14T02:23:35.000Z",
  "updated": "2015-01-22T21:20:50.000Z",
  "title": "Deriving the Qubit from Entropy Principles",
  "authors": [
    "Adam Brandenburger",
    "Pierfrancesco La Mura"
  ],
  "comment": "8 pages, 3 figures",
  "categories": [
    "quant-ph",
    "cs.IT",
    "math-ph",
    "math.IT",
    "math.MP"
  ],
  "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.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2014-07-09T03:36:04.000Z",
      "abstract": "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 maximum-entropy probabilities that satisfy an entropic version of the Heisenberg uncertainty principle. Our formulation is in phase space (following Wigner [1932]) and makes use of Renyi [1961] entropy (which includes Shannon [1948] 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.",
      "comment": null,
      "journal": null,
      "doi": null,
      "authors": [
        "Adam Brandenburger",
        "Matthew Deutsch",
        "Pierfrancesco La Mura"
      ]
    },
    {
      "version": "v5",
      "updated": "2015-01-22T21:20:50.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "heisenberg uncertainty principle",
      "phase space",
      "entropic version",
      "information reality principle says",
      "minimum entropy principle"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1311.3365B"
    }
  }
}