Quantum Foundations

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

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)

A new foundation of quantum mechanics for systems symmetric under a compact symmetry group is proposed. This is given by a link to classical statistics and coupled to the concept of a statistical parameter. A vector \phi of parameters is called an inaccessible c-variable if experiments can be provided for each single parameter, but no experiment can be provided for \phi. This is related to the concept of complementarity in quantum mechanics, but more generally to contrafactual parameters. Using these concepts and some weak assumptions, the Hilbert space of quantum mechanics is constructed. The complete set of axioms for time-independent quantum mechanics is provided by proving Born`s formula under weak assumptions. 1. Helland, I.S. (2006) Extended statistical modeling under symmetry: the link towards quantum mechanics. Ann. Statistics 34, 1, 42-77. 2. Helland, I.S. (2008) Quantum Mechanics from Focusing and Symmetry. Found. Phys. 38, 818-842. Reference: arXiv 0801.2026. (http://arxiv.org/abs/0801.2026)