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arXiv:1404.3921 · Physica Scripta

Negative probabilities and counter-factual reasoning in quantum cognition

J Acacio de Barros, Gary Oas

Published Dec 1, 2014• 35 citations
arXivHTMLPDF

Abstract

In this paper we discuss quantum-like decision-making experiments using negative probabilities. We do so by showing how the two-slit experiment, in the simplified version of the Mach–Zehnder interferometer, can be described by this formalism. We show that negative probabilities impose constraints to what types of counter-factual reasoning we can make with respect to (quantum) internal representations of the decision maker.

DOI
10.1088/0031-8949/2014/t163/014008

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In this paper we discuss quantum-like decision-making experiments using negative probabilities. We do so by showing how the two-slit experiment, in the simplified version of the Mach–Zehnder interferometer, can be described by this formalism. We show that negative probabilities impose constraints to what types of counter-factual reasoning we can make with respect to (quantum) internal representations of the decision maker.
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.
This article presents a general framework that allows irrational decision making to be theoretically investigated and simulated. Rationality in human decision making under uncertainty is normatively prescribed by the axioms of probability theory in order to maximize utility. However, substantial literature from psychology and cognitive science shows that human decisions regularly deviate from these axioms. Bistable probabilities are proposed as a principled and straight forward means for modeling (ir)rational decision making, which occurs when a decision maker is in "two minds". We show that bistable probabilities can be formalized by positive-operator-valued projections in quantum mechanics. We found that (1) irrational decision making necessarily involves a wider spectrum of causal relationships than rational decision making, (2) the accessible information turns out to be greater in irrational decision making when compared to rational decision making, and (3) irrational decision making is quantum-like because it violates the Bell-Wigner polytope.
We consider the following model of decision-making by cognitive systems. We present an algorithm -- quantum-like representation algorithm (QLRA) -- which provides a possibility to represent probabilistic data of any origin by complex probability amplitudes. Our conjecture is that cognitive systems developed the ability to use QLRA. They operate with complex probability amplitudes, mental wave functions. Since the mathematical formalism of QM describes as well (under some generalization) processing of such quantum-like (QL) mental states, the conventional quantum decision-making scheme can be used by the brain. We consider a modification of this scheme to describe decision-making in the presence of two ``incompatible'' mental variables. Such a QL decision-making can be used in situations like Prisoners Dilemma (PD) as well as others corresponding to so called disjunction effect in psychology and cognitive science.
Recently the mathematical formalism of quantum mechanics, especially methods of quantum probability theory, started to be widely used in a variety of applications outside of physics, e.g., cognition and psychology as well as economy and finances. To distinguish such models from genuine quantum physical models, they often called quantum-like (although often people simply speak about, e.g., "quantum cognition"). These novel applications generate a number of foundational questions. Nowadays we can speak about a new science - foundations of quantum-like modeling. At the first stage this science was mainly about comparison of classical and quantum models, mainly in the probabilistic setting. It was found that statistical data from cognitive psychology violate some basic constraints posed on data by classical probability theory (Kolmogorov, 1933); in particular, the constraints given by the formula of total probability and Bell's type inequalities. Recently another question attracted some attention. In spite of real success in applications, there are no reason to believe that the quantum probability would cover completely all problems of, e.g., cognition. May be more general probability models have to be explored. A similar problem attracted a lot of attention in foundations of quantum physics culminating in a series of experiments to check Sorkin's equality for the triple-slit experiment by Weihs' group. In this note we present a similar test in the cognitive experimental setting. Performance of this test would either give further confirmation of the adequacy of the quantum probability model to cognitive applications or rejection of the conventional quantum model. Thus this note opens the door for a series of exciting experimental tests for the quantum-like model of cognition.
This review paper has three main goals. First, to discuss a contextual neurophysiologically plausible model of neural oscillators that reproduces some of the features of quantum cognition. Second, to show that such a model predicts contextual situations where quantum cognition is inadequate. Third, to present an extended probability theory that not only can describe situations that are beyond quantum probability, but also provides an advantage in terms of contextual decision-making. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Engineering practice routinely involves decision making under uncertainty. Much of this decision making entails reconciling multiple pieces of information to form a suitable model of uncertainty. As more information is collected, one expectedly makes better and better decisions. However, conditional probability assessments made by human decision makers, as new information arrives does not always follow expected trends and instead exhibits inconsistencies. Understanding them is necessary for a better modeling of the cognitive processes taking place in their mind, whether it be the designer or the end-user. Doing so can result in better products and product features. Quantum probability has been used in the literature to explain many commonly observed deviations from the classical probability such as: question order effect, response replicability effect, Machina and Ellsberg paradoxes and the effect of positive and negative interference between events. In this work, we present results from a survey demonstrating responses that while difficult to explain using classical probability, can be explained using a quantum formulation - highlighting its potential in engineering applications. Since quantum formulism is more general and can also match the predictions of classical probability, it serves as a richer paradigm for modeling decision making behavior in engineering practice.