Talks

In a quantum-Bayesian delineation of quantum mechanics, the Born Rule cannot be interpreted as a rule for setting measurement-outcome probabilities from an objective quantum state. (A quantum system has potentially as many quantum states as there are agents considering it.) But what then is the role of the rule? In this paper, we argue that it should be seen as an empirical addition to Bayesian reasoning itself. Particularly, we show how to view the Born Rule as a normative rule in addition to usual Dutch-book coherence. It is a rule that takes into account how one should assign probabilities to the outcomes of various intended measurements on a physical system, but explicitly in terms of prior probabilities for and conditional probabilities consequent upon the imagined outcomes of a special counterfactual reference measurement. This interpretation is seen particularly clearly by representing quantum states in terms of probabilities for the outcomes of a fixed, fiducial symmetric informationally complete (SIC) measurement. We further explore the extent to which the general form of the new normative rule implies the full state-space structure of quantum mechanics. It seems to go some way.

I will discuss a set of strong, but probabilistically intelligible, axioms from which one can {\em almost} derive the appratus of finite dimensional quantum theory. These require that systems appear completely classical as restricted to a single measurement, that different measurements, and likewise different pure states, be equivalent up to the action of a compact group of symmetries, and that every state be the marginal of a bipartite state perfectly correlating two measurements. This much yields a mathematical representation of measurements, states and symmetries that is already very suggestive of quantum mechanics. One final postulate (a simple minimization principle, still in need of a clear interpretation) forces the theory's state space to be that of a formally real Jordan algebra

We review situations under which standard quantum adiabatic conditions fail. We reformulate the problem of adiabatic evolution as the problem of Hamiltonian eigenpath traversal, and give cost bounds in terms of the length of the eigenpath and the minimum energy gap of the Hamiltonians. We introduce a randomized evolution method that can be used to traverse the eigenpath and show that a standard adiabatic condition is recovered. We then describe more efficient methods for the same task and show that their implementation complexity is close to optimal.

Dr. Smolin is a faculty member at Perimeter Institute for Theoretical Physics and author of several books including most recently “The Trouble with Physics”. He is known for devising several different approaches to quantum gravity, in particular, loop quantum gravity. His research interests include cosmology, elementary particle theory, the foundations of quantum mechanics, and theoretical biology. This information Chalk and Talk will explore in special topics in Quantum Gravity.

Heisenberg Uncertainty Principle, Feynman sum over paths interpretation of quantum mechanics, and the harmonic oscillator: introduction to tunnelling, QFT (what is a photon?), and zero point energy.

In a 1960 paper, E. C. G. Stueckelberg showed how one can obtain the familiar complex-vector-space structure of quantum mechanics by starting with a real-vector-space theory and imposing a superselection rule. In this talk I interpret Stueckelberg’s construction in terms of a single auxiliary real-vector-space binary object—a universal rebit, or “ubit." The superselection rule appears as a limitation on our ability to measure the ubit or to use it in state transformations. This interpretation raises the following questions: (i) What is the ubit? (ii) Could the superselection rule emerge naturally as a result of decoherence? (iii) If so, could one hope to see experimentally any effects of imperfect decoherence? Background reading: E. C. G. Stueckelberg, "Quantum Theory in Real Hilbert Space," Helv. Phys. Acta 33, 727 (1960). P. Goyal, "From Information Geometry to Quantum Theory," arXiv:0805.2770. C. M. Caves, C. A. Fuchs, and P. Rungta, "Entanglement of Formation of an Arbitrary State of Two Rebits," arXiv:quant-ph/0009063.

Complex numbers are an intrinsic part of the mathematical formalism of quantum theory, and are perhaps its most mysterious feature. We show that it is possible to derive the complex nature of the quantum formalism directly from the assumption that a pair of real numbers is associated to each sequence of measurement outcomes, and that the probability of this sequence is a real-valued function of this number pair. By making use of elementary symmetry and consistency conditions, and without assuming that these real number pairs have any other algebraic structure, we show that these pairs must be manipulated according to the rules of complex arithmetic. We demonstrate that these complex numbers combine according to Feynman's sum and product rules, with the modulus-squared yielding the probability of a sequence of outcomes. Reference: arXiv:0907.0909 (http://arxiv.org/abs/0907.0909)