Session X33: Focus Session: Foundations of Quantum Theory

Prize for a Faculty Member for Research in an Undergraduate Institution Talk: Quantum Interference and Imaging with Correlated Photons

Advances in the production and detection of correlated photons in the last two decades have allowed numerous tests of the fundamentals of quantum mechanics and the implementation of quantum algorithms for quantum communication, computation and imaging. Polarization-entangled states of light have enabled many of these studies. The use spatial modes of light has opened new possibilities for entanglement and imaging, where the information resides in the correlation of photon pairs. Moreover, the accessibility of low-cost devices has enabled undergraduate labs that demonstrate fundamental predictions of quantum mechanics, such as superposition and nonlocality. It has also led to a movement to reinvent how we teach quantum mechanics in light of the rise of quantum information. In this talk I will present our research efforts in quantum information and imaging with correlated photons, and the use of these methods to implement undergraduate labs to teach quantum mechanics.   

Noncontextuality Inequalities

The Bell-Kochen-Specker theorem demonstrates that the predictions of quantum theory are inconsistent with a noncontextual hidden variable model. Significantly, the notion of noncontextuality to which it appeals is only well-defined for models of quantum theory as opposed to models of an arbitrary physical theory and then only for projective measurements and deterministic models thereof. By contrast, the notion of local causality introduced by Bell is not so restricted in its scope. In this talk, I present an operational definition of noncontextuality that recovers the traditional notion as a special case and allows one to define ``noncontextuality inequalities'' for experimental statistics. I will demonstrate that a particular two-party information-processing task, ``parity-oblivious multiplexing,'' is powered by contextuality in the sense that there is an inequality that bounds its performance in noncontextual models, and I will report on an experimental violation of this inequality in good agreement with the quantum predictions. Joint work with Daniel Buzacott, Tony Keehn, Ben Toner and Geoff Pryde.   

Quantum Bayesian Coherence

In the quantum-Bayesian development of quantum theory the Born Rule cannot be interpreted as a rule for setting measurement-outcome probabilities from an objective quantum state. But if not, what is the role of the rule? In this talk, 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 consequences 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 get quite far.   

Testing Quantum Mechanics using a Triple slit experiment

As one of the postulates of quantum mechanics, Born's rule tells us how to get probabilities for experimental outcomes from the complex wavefunction of the system. It's quadratic nature entails that interference occurs in pairs of paths. An experiment is in progress in our laboratory that sets out to test the correctness of Born's rule by testing for the presence or absence of genuine three-path interference [1]. This is done using single photons and a three slit aperture. Although the Born rule has been indirectly verified to high accuracy in other experiments, the consequences of a detection of even a small three-way interference in the Quantum mechanical null prediction are tremendous. If a non-zero result were to be obtained, it would mean that Quantum Mechanics is only approximate, in the same way that the double slit experiment proves that classical physics is only an approximation to the true law of nature. This would give us an important hint on how to generalize Quantum Mechanics and open a new window to the world. Some preliminary observations have been reported in reference [2]. In this talk, I will show results that bound the possible violation of the second sum rule and will point out ways to obtain a tighter experimental bound. [1] R. D. Sorkin, Mod. Phys. Lett. A 9, 3119 (1994). [2] U. Sinha et al, in Foundations of Probability and Physics-5, A I P Conference Proceedings, Vol. 1101, pp. 200-207, New-York (2009)   

Quantum State Spaces over Associative Composition Algebras

Wootters pointed out that a theory of quantum mechanics could be formulated without probability amplitudes [1]. Usual quantum theory over the complex field is among those in a hierarchy of theories indexed by their capacity, as defined in [2]. We explore the structure of quantum state spaces over associative composition algebras within the general Quantum Bayesian framework proposed by Fuchs and Schack [2]. We consider the possibility of expanding self-adjoint operators in terms of symmetric informationally complete bases for different algebraic modules. We chart the geometry of quantum state space on the corresponding probability simplexes by imposing a self- adjoint positive semi-definite nature to the pure states and their convex hull. \\[4pt] [1] W. K. Wootters, ``Quantum Mechanics without Probability Amplitudes,'' Foundations of Physics, Vol. 16, No. 4 (1985)\\[0pt] [2] C. A. Fuchs and R. Schack, ``Quantum-Bayesian Coherence,'' arXiv:0906.2187v1 [quant-ph] (2009)   

New Results on Kochen-Specker Setups

The Kochen-Specker (KS) theorem has recently been given renewed attention due to developments of both experimental and computational techniques as well as new theoretical results (more than 10 papers in Phys. Rev. Lett. in the last 10 years). We show that all possible KS setups for four-dimensional systems with 18 through 23 vectors and at least almost all with 24 vectors with component values from \{-1,0,1\} can be obtained by peeling vectors off a single system provided, in effect, by Asher Peres 20 years ago. We show that for 18 through 23 vectors there are no other such systems. We also show that there is number of systems with 18 through 24 vectors with component values that are not from \{-1,0,1\}. We present algorithms and computer programs that we used to generate all the aforementioned KS setups and we give their analytical, graphical, and statistical representation.   

Solving the mystery of wave/particle duality--the road to a unified theory of physics

The mystery of wave/particle duality persists because of the stubborn adherence to the point-particle model of elementary particles. This has excluded a whole class of theories based on a three-dimensional extended wave model. It is this class of theories which holds the promise of giving both an intuitively obvious resolution to the mystery of wave/particle duality and the key to the unification of the fundamental forces. It has been incredibly difficult, however, to construct a wave model that is consistent with the observed behavior of the objects we call elementary particles. We present here a new wave model which holds the promise of being just such a consistent model. This model gives an intuitively obvious explanation of wave/particle duality. Furthermore, this model opens up a new path in the search for a unified theory of elementary particles and the fundamental forces.   

Elastic Rotational Solitons as Elementary Particles

By assuming a linear response to variations of orientation in an ideal isotropic elastic solid, we derive a nonlinear Dirac equation which describes rotational waves. This result provides a simple mechanical interpretation of relativistic quantum mechanical dynamics. The energy, momentum, and angular momentum operators are derived. Fermion and boson solutions may both be possible. Correlations between states have the quantum mechanical form. Half-integer spin arises from the fact that waves propagating in opposite directions form independent states 180 degrees apart. The Pauli exclusion principle and interaction potentials are derived from the assumption of independent interacting soliton ``particles.''   

Finite Quantum Electrodynamics

In quantum interactions theory, each intermediate particle contributes to the scattering amplitude a factor inversely proportional to the deviation of the intermediate particle from its real state. We apply this principle by considering that the masses of the intermediate particles differ from the masses of the real ones and obtain electromagnetic scattering amplitudes free from ultraviolet divergences. Furthermore, this method has the flexibility of alternative approaches to other kinds of interactions.   

Classical Derivation of Planck's Radiation Law

Inverse scaling between energy and wavelength renders the expression $h$=\textit{E$\lambda $/c} constant, and a formal identity. As a mathematical consequence, $h$ is constant for a continuum of energy and associated wavelength values. The \textit{E$\lambda $/c=Const}. identity entails the possibility that $h$ is expression and universal manifestation of the fact that $E\lambda $ product of energy and spatial interval, as well as $E\Delta \tau _{ }$product of energy and temporal interval of photons in four-space are invariants. Assumed to be so, derivation of the spectral radiance of black-body radiation may not be fundamentally conditioned on quantum discontinuity, per se. As apparent vindication of such eventuality, we present a first of a kind derivation of Planck's radiation law, completely within the framework of classical physics, and without need to invoke Planck's quantum of action or quantum discontinuity. An accordingly augmented notion of Planck's constant enables a likewise classical derivation of the quantum Hall impedance, the magnetic flux quantum and the fine structure constant. It is shown that the numerical value of $h,$ the quantum Hall impedance, the magnetic flux quantum and the fine structure constant can be calculated from three fundamental physical quantities; the speed of light, the electron charge, and the wavelength of unit-energy (1eV) photon.   

Inside and Outside in Four Dimensions

The author produces a novel proof that Inside and Outside are in fact directions in four space. It has long been suspected that an escape from a three dimensional container would involve an extra dimension. This idea of inside being the opposite of outside can also be used in Quantum Computation to construct an algorithm that can be used to solve boundary value problems, where a trapped object can be released from a closed contatiner through the use of an extra dimension. This algorithm has uses in intractable problems and otherwise insolvable problems involving objects trapped by boundaries.