Talks

Recent advances in quantum computation and quantum information theory have led to revived interest in, and cross-fertilisation with, foundational issues of quantum theory. In particular, it has become apparent that quantum theory may be interpreted as but a variant of the classical theory of probability and information. While the two theories may at first sight appear widely different, they actually share a substantial core of common properties; and their divergence can be reduced to a single attribute only, their respective degree of agent-dependency. I propose a mathematical description for this ?degree of agent-dependency? and show how assuming different values allows one to derive the classical and the quantum case from their common core. Finally, I explore ? and eventually dismiss ? the possibility that beyond quantum theory there might be other variants of classical probability theory that are relevant to physics.

The starting point of the reconstruction process is a very simple quantum logical structure on which probability measures (states) and conditional probabilities are defined. This is a generalization of Kolmogorov's measure-theoretic approach to probability theory. In the general framework, the conditional probabilities need neither exist nor be uniquely determined if they exist. Postulating their existence and uniqueness becomes the major step in the reconstruction process. A certain new mathematical structure can then be derived, and examples immediately reveal that probability conditionalization is identical with the Lüders - von Neumann measurement process. Some further postulates bring us to Jordan algebras, and the consideration of composite systems finally shows why these algebras must be the self-adjoint parts of von Neumann algbras such that they can be represented as linear operators on Hilbert spaces over the complex numbers. This is why the approach gets ahead of other ones that are not able to justify the need for the complex Hilbert space or the Jordan operator algebras. The mathematical structure of quantum mechanics can thus be reconstructed from a few probabilistic basic principles and becomes a non-Boolean extension of classical probability theory. Its link to physics is that probability conditionalization in this structure is identical with the Lüders - von Neumann measurement process.

In our approach, rather than aiming to recover the 'Hilbert space model' which underpins the orthodox quantum mechanical formalism, we start from a general `pre-operational' framework, and verify how much additional structure we need to be able to describe a range of quantum phenomena. This also enables us to investigate which mathematical models, including more abstract categorical ones, enable one to model quantum theory. Till now, all of our axioms only refer to the particular nature of how compound quantum systems interact, rather that to the particular structure of state-spaces. This is in sharp contrast with other approaches of this kind which aim to recover quantum theory out of a much broader class of theories. A more abstract quantum mechanical model has other many advantages. It elucidates which are the key ingredients that make `the Hilbert space model' work. Since it relies on monoidal categories, it comes with a high-level diagrammatic description (which we think of as `the' mathematical formalism). It moreover removes the dependency on continuous underlying mathematical structures, paving the way for discrete combinatorial models, which might blend better with the other ingredients required for a theory of quantum gravity.

It will be shown that the conventional (i.e. real or complex Hilbert space) model of quantum mechanics  can be deduced from the indistinguishability of the simplest types of statistical mixtures. The result does not have the  low dimension exclusion of the  quantum logic approach.

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.