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arXiv:1412.4888

Negative probabilities and Counterfactual Reasoning on the double-slit Experiment

J. Acácio de Barros, Gary Oas, Patrick Suppes

Published Dec 16, 2014• 14 citations

Abstract

In this paper we attempt to establish a theory of negative (quasi) probability distributions from fundamental principles and apply it to the study of the double-slit experiment in quantum mechanics. We do so in a way that preserves the main conceptual issues intact but allow for a clearer analysis, by representing the double-slit experiment in terms of the Mach-Zehnder interferometer, and show that the main features of quantum systems relevant to the double-slit are present also in the Mach-Zehnder. This converts the problem from a continuous to a discrete random variable representation. We then show that, for the Mach-Zehnder interferometer, negative probabilities do not exist that are consistent with interference and which-path information, contrary to what Feynman believed. However, consistent with Scully et al.'s experiment, if we reduce the amount of experimental information about the system and rely on counterfactual reasoning, a joint negative probability distribution can be constructed for the Mach-Zehnder experiment.

DOI: 10.48550/arxiv.1412.4888

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In this paper we attempt to establish a theory of negative (quasi) probability distributions from fundamental principles and apply it to the study of the double-slit experiment in quantum mechanics. We do so in a way that preserves the main conceptual issues intact but allow for a clearer analysis, by representing the double-slit experiment in terms of the Mach-Zehnder interferometer, and show that the main features of quantum systems relevant to the double-slit are present also in the Mach-Zehnder. This converts the problem from a continuous to a discrete random variable representation. We then show that, for the Mach-Zehnder interferometer, negative probabilities do not exist that are consistent with interference and which-path information, contrary to what Feynman believed. However, consistent with Scully et al.'s experiment, if we reduce the amount of experimental information about the system and rely on counterfactual reasoning, a joint negative probability distribution can be constructed for the Mach-Zehnder experiment.

Abstract Quantum mechanics is a physical theory based on statistical laws and can merely predict probabilities for the measurement outcomes, suggesting that it is an average description. Due to the lack of all the details of microscopic physical objects, only probabilistic predictions can be made for physical phenomena. In this paper, we discuss the root of the probabilistic descriptions in quantum theory by recalling the single photon interference experiments conducted by two commonly used optical interferometric systems: the Mazh–Zehnder interferometer (MZI) and Young’s double-slit system (YDS). Though they are both two-path interference, it is shown that the quantum mechanics can provide deterministic predictions for the measurement outcomes in some special cases by adopting the MZI, while only probabilistic predictions can be provided for the measurement outcomes by adopting the YDS. We hope this work can help to understand the probabilistic nature of quantum mechanics.

After reviewing basic concepts and tools in order to quantify entanglement, a theoretical set up is proposed, which is based on the double slit experiment and the Mach-Zehnder interferometer. A particle, which is in a superposition of two paths, can interfere with a qubit on each path before it is scattered by a beam splitter and measured by two separate detectors. After a detailed investigation, it can be shown that it is possible to induce entanglement to the qubits via post-selection. This means that the measurement does not induce entanglement, but can be used to select the entangled qubits after a number of iterations. A remarkable result is, that this works independently from the probability of interaction between the particle and the qubits, which could turn out be very beneficial in a possible realization of the set up.

This paper intends to clarify some issues in the theory of quantum measurement by taking advantage of the self-consistent quantum formulation of nonlinear optics. A quantum-nondemolition measurement of the photon number of an optical pulse can be performed with a nonlinear Mach-Zehnder interferometer followed by a balanced detector. The full quantum-mechanical treatment shows that the shortcut in the description of the quantum-mechanical measurement, the so-called ``collapse of the wave function,'' is not needed for a self-consistent interpretation of the measurement process. Coherence in the density matrix of the signal to be measured is progressively reduced with increasing accuracy of the photon-number determination. The quantum-nondemolition measurement is incorporated in the double-slit experiment and the contrast ratio of the fringes is found to decrease systematically with increasing information on the photon number in one of the two paths. The ``gain'' in the measurement can be made arbitrarily large so that postprocessing of the information can proceed classically.