Abstract:
Counterfactual definiteness must be used as at least one of the postulates or axioms that are necessary to derive Bell-type inequalities. It is considered by many to be a postulate that not only is commensurate with classical physics (as for example Einstein’s special relativity), but also separates and distinguishes classical physics from quantum mechanics. It is the purpose of this paper to show that Bell’s choice of mathematical functions and independent variables implicitly includes counterfactual definiteness. However, his particular choice of variables reduces the generality of his theory, as well as the physics of all Bell-type theories, so significantly that no meaningful comparison of these theories with actual Einstein-Podolsky-Rosen experiments can be made.

Abstract:
We present a detailed analysis of the set theoretical proof of Wigner for Bell type inequalities with the following result. Wigner introduced a crucial assumption that is not related to Einstein’s local realism, but instead, without justification, to the existence of certain joint probability measures for possible and actual measurement outcomes of Einstein-Podolsky-Rosen (EPR) experiments. His conclusions about Einstein’s local realism are, therefore, not applicable to EPR experiments and the contradiction of the experimental outcomes to Wigner’s results has no bearing on the validity of Einstein’s local realism.

Abstract:
A discrete-event simulation approach which provides a cause-and-effect description of many experiments with photons and neutrons exhibiting interference and entanglement is applied to a recent single-neutron experiment that tests (generalizations of) Heisenberg's uncertainty relation. The event-based simulation algorithm reproduces the results of the quantum theoretical description of the experiment but does not require the knowledge of the solution of a wave equation nor does it rely on concepts of quantum theory. In particular, the data satisfies uncertainty relations derived in the context of quantum theory.

Abstract:
We construct an event-based computer simulation model of the Einstein-Podolsky-Rosen-Bohm experiments with photons. The algorithm is a one-to-one copy of the data gathering and analysis procedures used in real laboratory experiments. We consider two types of experiments, those with a source emitting photons with opposite but otherwise unpredictable polarization and those with a source emitting photons with fixed polarization. In the simulation, the choice of the direction of polarization measurement for each detection event is arbitrary. We use three different procedures to identify pairs of photons and compute the frequency of coincidences by analyzing experimental data and simulation data. The model strictly satisfies Einstein's criteria of local causality, does not rely on any concept of quantum theory and reproduces the results of quantum theory for both types of experiments. We give a rigorous proof that the probabilistic description of the simulation model yields the quantum theoretical expressions for the single- and two-particle expectation values.

Abstract:
We address the basic meaning of apparent contradictions of quantum theory and probability frameworks as expressed by Bell's inequalities. We show that these contradictions have their origin in the incomplete considerations of the premisses of the derivation of the inequalities. A careful consideration of past work, including that of Boole and Vorob'ev, has lead us to the formulation of extended Boole-Bell inequalities that are binding for both classical and quantum models. The Einstein-Podolsky-Rosen-Bohm gedanken experiment and a macroscopic quantum coherence experiment proposed by Leggett and Garg are both shown to obey the extended Boole-Bell inequalities. These examples as well as additional discussions also provide reasons for apparent violations of these inequalities.

Abstract:
In their comment[1] on our Letter [arXiv:0907.0767], Leggett and Garg claim that they have introduced in their original paper (LG1) a dependence on measurement times. They also claim that Eqs.(HMDR1) and (LG2a) can therefore not be linked in such a way that the arguments of [arXiv:0907.0767] can be transcribed. However, (LG1) distinguishes only three time differences, and all experimental results corresponding to the same time differences are identically labeled and therefore treated as mathematically identical. We therefore cannot agree with the argumentation of Leggett and Garg: except for a change of nomenclature Eqs.(HMDR1) and (LG2a) are the same. A more extensive discussion of this point can be found in [arXiv:0901.2546].

Abstract:
It is demonstrated that the third-order interference, as obtained from explicit solutions of Maxwell's equations for realistic models of three-slit devices, including an idealized version of the three-slit device used in a recent three-slit experiment with light (U. Sinha et al., Science 329, 418 (2010)), is generally nonzero. The hypothesis that the third-order interference should be zero is shown to be fatally flawed because it requires dropping the one-to-one correspondence between the symbols in the mathematical theory and the different experimental configurations.

Abstract:
We describe an event-based approach to simulate the propagation of an electromagnetic plane wave through dielectric media. The basic building block is a deterministic learning machine that is able to simulate a plane interface. We show that a network of two of such machines can simulate the propagation of light through a plane parallel plate. With properly chosen parameters this setup can be used as a beam splitter. The modularity of the simulation method is illustrated by constructing a Mach-Zehnder interferometer from plane parallel plates, the whole system reproducing the results of wave theory. A generalization of the event-based model of the plane parallel plate is also used to simulate a periodically stratified medium.

Abstract:
A discrete-event approach, which has already been shown to give a cause-and-effect explanation of many quantum optics experiments, is applied to single-neutron interferometry experiments. The simulation algorithm yields a logically consistent description in terms of individual neutrons and does not require the knowledge of the solution of a wave equation. It is shown that the simulation method reproduces the results of several single-neutron interferometry experiments, including experiments which, in quantum theoretical language, involve entanglement. Our results demonstrate that classical (non-Hamiltonian) systems can exhibit correlations which in quantum theory are associated with interference and entanglement, also when all particles emitted by the source are accounted for.

Abstract:
The quantum delayed-choice experiment of Tang et al. [Nature Photonics 6 (2012) 600] is simulated on the level of individual events without making reference to concepts of quantum theory or without solving a wave equation. The simulation results are in excellent agreement with the quantum theoretical predictions of this experiment. The implication of the work presented in the present paper is that the experiment of Tang et al. can be explained in terms of cause-and-effect processes in an event-by-event manner.