FOUNDATIONS OF PROBABILITY AND PHYSICS  3

Some loopholes to save quantum nonlocality
View Description Hide DescriptionThe EPR‐chameleon experiment has closed a long standing debate between the supporters of quantum nonlocality and the thesis of quantum probability according to which the essence of the quantum pecularity is non Kolmogorovianity rather than non locality.
The theory of adaptive systems (symbolized by the chameleon effect) provides a natural intuition for the emergence of non‐Kolmogorovian statistics from classical deterministic dynamical systems. These developments are quickly reviewed and in conclusion some comments are introduced on recent attempts to “reconstruct history” on the lines described by Orwell in “1984”.

The Quantum Stalemate
View Description Hide DescriptionThe purpose of this paper is to advocate for more thorough investigations of EPR experiments with photons, and in particular of the main remaining loophole (i.e., the low detection efficiency loophole) using some recent proposals to circumvent this problem.

The quantum measurement process in an exactly solvable model
View Description Hide DescriptionAn exactly solvable model for a quantum measurement is discussed which is governed by hamiltonian quantum dynamics. The z‐component ŝ_{z} of a spin is measured with an apparatus, which itself consists of magnet coupled to a bath. The initial state of the magnet is a metastable paramagnet, while the bath starts in a thermal, gibbsian state. Conditions are such that the act of measurement drives the magnet in the up or down ferromagnetic state according to the sign of s_{z} of the tested spin. The quantum measurement goes in two steps. On a timescale the off‐diagonal elements of the spin’s density matrix vanish due to a unitary evolution of the tested spin and the N apparatus spins; on a larger but still short timescale this is made definite by the bath. Then the system is in a ‘classical’ state, having a diagonal density matrix. The registration of that state is a quantum process which can already be understood from classical statistical mechanics. The von Neumann collapse and the Born rule are derived rather than postulated.

Single‐photon generation and simultaneous observation of wave and particle properties
View Description Hide DescriptionWe describe an experiment in that generates single photons on demand and measures properties accounted to both particle and wave‐like features of light. The measurement is performed by exploiting data that are sampled simultaneously in a single experimental run.

Quantum Tomography and Verification of Generalized Bell‐CHSH Inequalities
View Description Hide DescriptionIt is shown that verification procedure of Bell inequalities consists of three parts. The first stage of verification procedure is consist of measurements of a set of spin correlations. Using them one can reconstruct a density matrix of two‐particle state. It can be done with the help of quantum tomography technique. At the second stage one must construct a generalized Bell inequality corresponding to this state. This inequality must be constructed in two forms: within the framework of the traditional formulation of quantum mechanics and in the frames of a local hidden variables theory. Only after that the results of experimental measurements can be compared with theoretical predictions. It is a third stage of verification.

Epistemic and Ontic Quantum Realities
View Description Hide DescriptionQuantum theory has provoked intense discussions about its interpretation since its pioneer days, beginning with Bohr’s view of quantum theory as a theory of knowledge. We show that such an epistemic perspective can be consistently complemented by Einstein’s ontically oriented position.

MUBs, Polytopes, and Finite Geometries
View Description Hide DescriptionA complete set of N + 1 mutually unbiased bases (MUBs) exists in Hilbert spaces of dimension N = p^{k} , where p is a prime number. They mesh naturally with finite affine planes of order N, that exist when N = p^{k} . The existence of MUBs for other values of N is an open question, and the same is true for finite affine planes. I explore the question whether the existence of complete sets of MUBs is directly related to the existence of finite affine planes. Both questions can be shown to be geometrical questions about a convex polytope, but not in any obvious way the same question.

On the vector Helmholtz equation in toroidal waveguides
View Description Hide DescriptionA wave splitting method is proposed to solve the problem of propagation of microwaves in a circular waveguide bend of circular cross section. The splitting method, applied to the vector Helmholtz equation, gives a stable solution in terms of waves propagating to the right and to the left in the bend. The formulation is particularly transparent for analyzing the scattering properties of toroidal bends. The basis for the transparency of the method is that the wave splitting is formally exact as the exponential of the square root of a differential operator. The modal functions of the straight cylindrical waveguide are chosen as basis functions in the transverse quasi‐toroidal variables.

Not What Models Are, But What Models Do
View Description Hide DescriptionA distinction between two types of models is proposed based on the articulated recognition that different models are brought to play different roles in the process of theory‐building.

Tracing the bounds on Bell‐type inequalities
View Description Hide DescriptionBell‐type inequalities and violations thereof reveal the fundamental differences between standard probability theory and its quantum counterpart. In the course of previous investigations ultimate bounds on quantum mechanical violations have been found. For example, Tsirelson’s bound constitutes a global upper limit for quantum violations of the Clauser‐Horne‐Shimony‐Holt (CHSH) and the Clauser‐Horne (CH) inequalities. Here we investigate a method for calculating the precise quantum bounds on arbitrary Bell‐type inequalities by solving the eigenvalue problem for the operator associated with each Bell‐type inequality. Thereby, we use the min‐max principle to calculate the norm of these self‐adjoint operators from the maximal eigenvalue yielding the upper bound for a particular set of measurement parameters. The eigenvectors corresponding to the maximal eigenvalues provide the quantum state for which a Bell‐type inequality is maximally violated.

Quantum Filtering Theory and the Filtering Interpretation
View Description Hide DescriptionStarting from an elementary level we present the theory of quantum filtering. The physical and philosophical principles of stochastic master equations describing the dynamics of a continuously observed quantum system are explained in terms of quantum filtering equations. We believe these principles constitute a consistent interpretation of quantum theory and measurement.

A Non‐Intuitionist’s Approach To The Interpretation Problem Of Quantum Mechanics
View Description Hide DescriptionA philosophy of physics called “linguistic empiricism” is presented and applied to the interpretation problem of quantum mechanics. This philosophical position is based on the works of Jacques Derrida. The main propositions are (i) that meaning, included the meaning attached to observations, are language‐dependent and (ii) that mathematics in physics should be considered as a proper language, not necessary translatable to a more basic language of intuition and immediate experience. This has fundamental implications for quantum mechanics, which is a mathematically coherent and consistent theory; its interpretation problem is associated with its lack of physical images expressible in ordinary language.

Distributivity breaking and macroscopic quantum games
View Description Hide DescriptionExamples of games between two partners with mixed strategies, calculated by the use of the probability amplitude as some vector in Hilbert space are given. The games are macroscopic, no microscopic quantum agent is supposed. The reason for the use of the quantum formalism is in breaking of the distributivity property for the lattice of yes‐no questions arising due to the special rules of games. The rules of the games suppose two parts: the preparation and measurement. In the first part due to use of the quantum logical orthocomplemented non‐distributive lattice the partners freely choose the wave functions as descriptions of their strategies. The second part consists of classical games described by Boolean sublattices of the initial non‐Boolean lattice with same strategies which were chosen in the first part. Examples of games for spin one half are given. New Nash equilibria are found for some cases. Heisenberg uncertainty relations without the Planck constant are written for the “spin one half game”.

Fuzzy Quantum Probability Theory
View Description Hide DescriptionThis paper surveys some of the recent results that have been obtained in fuzzy quantum probability theory. In this theory fuzzy events are represented by quantum effects and fuzzy random variables are represented by quantum measurements. Probabilities, conditional probabilities and sequential products of effects are discussed. The relationship between the law of total probability and compatibility of measurements is treated. Results concerning independent effects are given. Finally, properties of the sequential product, almost sharp effects and nearly sharp effects are discussed.

Fuzzy position and momentum observables
View Description Hide DescriptionThe notion of a fuzzy observable is a possible way to describe imprecise measurements. We show that fuzzy versions of the usual position and momentum observables are exactly those observables which satisfy the natural covariance and invariance requirements. We characterize the class of fuzzy position and momentum observables which are informationally equivalent with the canonical position and momentum observables.

Quantum theory as a statistical theory under symmetry
View Description Hide DescriptionBoth statistics and quantum theory deal with prediction using probability. We will show that there can be established a connection between these two areas. This will at the same time suggest a new, less formalistic way of looking upon basic quantum theory.
A total parameter space Φ, equipped with a group G of transformations, gives the mental image of some quantum system, in such a way that only certain components, functions of the total parameter φ can be estimated. Choose an experiment/question a, and get from this a parameter space Λ^{ a }, perhaps after some model reduction compatible with the group structure. As in statistics, it is important always to distinguish between observations and parameters, in particular between (minimal) observations t^{a} and state variables (parameters) λ^{ a }. Let K ^{ a } be the L^{2} ‐space of functions of t^{a} , and let H ^{ a } be the image of K ^{ a } under the expectation operator of the model. The measure determining the L^{2}‐spaces is the invariant measure under the maximal subgroup which induces a transformation on Λ^{ a }.
There is a unitary connection between K ^{ a } and H ^{ a }, and then under natural conditions between H ^{ a } and H ^{ b } for a and b are different. Thus there exists a common Hilbert space H such that H ^{ a } equals U^{a} H for a unitary U^{a} . In agreement with the common formulation of quantum mechanics, the vectors of H are taken to represent the states of the system. The state interpretation is then: An ideal experiment E ^{ a } has been performed, and the result of this experiment, disregarding measurement errors, is that the parameter λ^{ a } is equal to some fixed number . This essentially statistical construction leads under natural assumptions to the basic axioms of quantum mechanics, and thus implies a new statistical interpretation of this traditionally very formal theory. The probabilities are introduced via Born’s formula, and this formula is proved from general, reasonable assumptions, essentially symmetry assumptions.
The theory is illustrated by a simple macroscopic example, and by the example of a spin particle. As a last example we show a connection to inference between related macroscropic experiments under symmetry. The result of this last example seems to have potential for being generalized considerably.

Bell’s theorem: Critique of proofs with and without inequalities
View Description Hide DescriptionMost of the standard proofs of the Bell theorem are based on the Kolmogorov axioms of probability theory. We show that these proofs contain mathematical steps that cannot be reconciled with the Kolmogorov axioms. Specifically we demonstrate that these proofs ignore the conclusion of a theorem of Vorob’ev on the consistency of joint distributions. As a consequence Bell’s theorem stated in its full generality remains unproven, in particular, for extended parameter spaces that are still objective local and that include instrument parameters that are correlated by both time and instrument settings. Although the Bell theorem correctly rules out certain small classes of hidden variables, for these extended parameter spaces the standard proofs come to a halt. The Greenberger‐Horne‐Zeilinger (GHZ) approach is based on similar fallacious arguments. For this case we are able to present an objective local computer experiment that simulates the experimental test of GHZ performed by Pan, Bouwmeester, Daniell, Weinfurter and Zeilinger and that directly contradicts their claim that Einstein‐local elements of reality can neither explain the results of quantum mechanical theory nor their experimental results.

Projection scheme for a reflected stochastic heat equation with additive noise
View Description Hide DescriptionWe consider a projection scheme as a numerical solution of a reflected stochastic heat equation driven by a space‐time white noise. Convergence is obtained via a discrete contraction principle and known convergence results for numerical solutions of parabolic variational inequalities.

Degenerate Diffusions with regular Hamiltonians
View Description Hide DescriptionWe consider the asymptotic behaviour for small time and small parameter of the heat kernel on the cotangent bundle of a Riemannian manifold defined by a stochastically perturbed geodesic flow generalizing WKB‐type methods to a diffusion corresponding to a particular regular degenerate Hamiltonian. For small time equivalence of the dynamics for the degenerate Hamiltonian system and for the system given by the corresponding Hamilton‐Jacobi equation is used.