Quantum Foundations

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)

The quantum equations for bosonic fields may be derived using an 'exact uncertainty' approach [1]. This method of quantization can be applied to fields with Hamiltonian functionals that are quadratic in the momentum density, such as the electromagnetic and gravitational fields. The approach, when applied to gravity [2], may be described as a Hamilton-Jacobi quantization of the gravitational field. It differs from previous approaches that take the classical Hamilton-Jacobi equation as their starting point in that it incorporates some new elements, in particular the use of a formalism of ensembles on configuration space and the postulate of an exact uncertainty relation. These provide the fundamental elements needed for the transition to the quantum theory. The formalism of ensembles on configuration space is general enough to describe classical, quantum, and interacting classical-quantum systems in a consistent way. This is of some relevance to gravity: although there are many physical arguments in favour of a quantum theory of gravity, it appears that the justification for such a theory does not follow from logical arguments alone [3]. It is therefore of interest to consider the coupling of quantum fields to a classical gravitational field. This leads to a theory that is fundamentally different from standard semiclassical gravity. 1. Michael J W Hall, Kailash Kumar and Marcel Reginatto, Bosonic field equations from an exact uncertainty principle, J. Phys. A 36 (2003) 9779-9794 (http://arxiv.org/abs/hep-th/0307259). 2. M. Reginatto, Exact Uncertainty Principle and Quantization: Implications for the Gravitational Field, Proceedings of DICE2004 in: Braz. J. Phys. 35 (2005) 476-480 (http://arxiv.org/abs/gr-qc/0501030). 3. Mark Albers, Claus Kiefer and Marcel Reginatto, Measurement analysis and quantum gravity, Phys. Rev. D 78 (2008) 064051 (http://arxiv.org/abs/0802.1978)

The fact that quantum mechanics admits exact uncertainty relations is used to motivate an ‘exact uncertainty’ approach to obtaining the Schrödinger equation. In this approach it is assumed that an ensemble of classical particles is subject to momentum fluctuations, with the strength of the fluctuations determined by the classical probability density [1]. The approach may be applied to any classical system for which the Hamiltonian is quadratic with respect to the momentum, including all physical particles and fields [2]. The approach is based on a general formalism that describes physical ensembles via a probability density P on configuration space, together with a canonically conjugate quantity S [3]. Quantum and classical ensembles are particular cases of interest, but one can also ask more general questions within this formalism, such as (i) Can one consistently describe interactions between quantum and classical systems? and (ii) Can one obtain local nonlinear modifications of quantum mechanics? These questions will be briefly discussed, with respect to measurement interactions and spin-1/2 systems respectively. 1. M.J.W. Hall and M. Reginatto, “Schroedinger equation from an exact uncertainty principle”, J. Phys. A 35 (2002) 3289 (http://lanl.arxiv.org/abs/quant-ph/0102069). 2. M.J.W. Hall, “Exact uncertainty approach in quantum mechanics and quantum gravity”, Gen. Relativ. Gravit. 37 (2005) 1505 (http://lanl.arxiv.org/abs/gr-qc/0408098). 3. M.J.W. Hall and M. Reginatto, “Interacting classical and quantum systems”, Phys. Rev. A 72 (2005) 062109 (http://lanl.arxiv.org/abs/quant-ph/0509134).

Non-relativistic quantum theory is derived from information codified into an appropriate statistical model. The basic assumption is that there is an irreducible uncertainty in the location of particles so that the configuration space is a statistical manifold with a natural information metric. The dynamics then follows from a principle of inference, the method of Maximum Entropy: entropic dynamics is an instance of law without law. The concept of time is introduced as a convenient device to keep track of the accumulation of changes. The resulting formalism is close to Nelson's stochastic mechanics. The statistical manifold is a dynamical entity: its (information) geometry determines the evolution of the probability distribution which, in its turn, reacts back and determines the evolution of the geometry. As in General Relativity there is a kind of equivalence principle in that “fictitious” forces – in this case diffusive “osmotic” forces – turn out to be “real”. This equivalence of quantum and statistical fluctuations – or of quantum and classical probabilities – leads to a natural explanation of the equality of inertial and “osmotic” masses and allows explaining quantum theory as a sophisticated example of entropic inference. Mass and the phase of the wave function are explained as features of purely statistical origin. Recommended Reading: arXiv:0907.4335 "From Entropic Dynamics to Quantum Theory" (2009)

I will consider physical theories which describe systems with limited information content. This limit is not due observer's ignorance about some “hidden” properties of the system - the view that would have to be confronted with Bell's theorem - but is of fundamental nature. I will show how the mathematical structure of these theories can be reconstructed from a set of reasonable axioms about probabilities for measurement outcomes. Among others these include the “locality” assumption according to which the global state of a composite system is completely determined by correlations between local measurements. I will demonstrate that quantum mechanics is the only theory from the set in which composite systems can be in entangled (non-separable) states. Within Hardy's approach this feature allows to single out quantum theory from other probabilistic theories without a need to assume the “simplicity” axiom. 1. Borivoje Dakic, Caslav Brukner (in preparation) 2. Caslav Brukner, Anton Zeilinger, Information Invariance and Quantum Probabilities, arXiv:0905.0653 3. Tomasz Paterek, Borivoje Dakic, Caslav Brukner, Theories of systems with limited information content, arXiv:0804.1423