Gary Oas

Negative probabilities and counter-factual reasoning in quantum cognition

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.

Measuring Observable Quantum Contextuality

Contextuality is a central property in comparative analysis of classical, quantum, and supercorrelated systems. We examine and compare two well-motivated approaches to contextuality. One approach (“contextuality-by-default”) is based on the idea that one and the same physical property measured under different conditions (contexts) is represented by different random variables. The other approach is based on the idea that while a physical property is represented by a single random variable irrespective of its context, the joint distributions of the random variables describing the system can involve negative (quasi-)probabilities. We show that in the Leggett-Garg and EPR-Bell systems, the two measures essentially coincide.

Exploring non-signalling polytopes with negative probability

Bipartite and tripartite EPR–Bell type systems are examined via joint quasi-probability distributions where probabilities are permitted to be negative. It is shown that such distributions exist only when the no-signalling condition is satisfied. A characteristic measure, the probability mass, is introduced and, via its minimization, limits the number of quasi-distributions describing a given marginal probability distribution. The minimized probability mass is shown to be an alternative way to characterize non-local systems. Non-signalling polytopes for two to eight settings in the bipartite scenario are examined and compared to prior work. Examining perfect cloning of non-local systems within the tripartite scenario suggests defining two categories of signalling. It is seen that many properties of non-local systems can be efficiently described by quasi-probability theory.

Universal cubic eigenvalue repulsion for random normal matrices

Random matrix models consisting of normal matrices, defined by the sole constraint

Some Examples of Contextuality in Physics: Implications to Quantum Cognition

[N^{\dag},N]=0

Quantum Cognition, Neural Oscillators, and Negative Probabilities

, will be explored. It is shown that cubic eigenvalue repulsion in the complex plane is universal with respect to the probability distribution of matrices. The density of eigenvalues, all correlation functions, and level spacing statistics are calculated. Normal matrix models offer more probability distributions amenable to analytical analysis than complex matrix models where only a model wth a Gaussian distribution are solvable. The statistics of numerically generated eigenvalues from gaussian distributed normal matrices are compared to the analytical results obtained and agreement is seen.

A Survey of Physical Principles Attempting to Define Quantum Mechanics

Contextuality, the impossibility of assigning a single random variable to represent the outcomes of the same measurement procedure under different experimental conditions, is a central aspect of quantum mechanics. Thus defined, it appears in well-known cases in quantum mechanics, such as the double-slit experiment, the Bell-EPR experiment, and the Kochen-Specker theorem. Here we examine contextuality in such cases, and discuss how each of them bring different conceptual issues when applied to quantum cognition. We then focus on the shortcomings of using quantum probabilities to describe social systems, and explain how negative quasi-probability distributions may address such limitations.

Mapping Quantum Reality: What to Do When the Territory Does Not Make Sense?

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.

Phase-oscillator computations as neural models of stimulus–response conditioning and response selection

Quantum mechanics, one of the most successful theories in the history of science, was created to account for physical systems not describable by classical physics. Though it is consistent with all experiments conducted thus far, many of its core concepts (amplitudes, global phases, etc.) can not be directly accessed and its interpretation is still the subject of intense debate, more than 100 years since it was introduced. So, a fundamental question is why this particular mathematical model is the one that nature chooses, if indeed it is the correct model. In the past two decades there has been a renewed effort to determine what physical or informational principles define quantum mechanics. In this paper, recent attempts at establishing reasonable physical principles are reviewed and their degree of success is tabulated. An alternative approach using joint quasi-probability distributions is shown to provide a common basis of representing most of the proposed principles. It is argued that having a common representation of the principles can provide intuition and guidance to relate current principles or advance new principles. The current state of affairs, along with some alternative views are discussed.

A Collection of Probabilistic Hidden-Variable Theorems and Counterexamples

One of the central goals of science is to find consistent and rational representations of observational data: a map of the world, if you will. How we do this depends on the specific tools, which are often mathematical. When dealing with real-world situations, where the data (or “territory”) has a random component, the mathematical tools most commonly used are those grounded in probability theory, defined in a precise way by the Russian mathematician Andrei Kolmogorov. In this paper we explore how experimental data (the “territory”) can be represented (or “mapped”) consistently in terms of probability theory, and present examples of situations, both in the physical and social sciences, where such representations are impossible. This suggests that some “territories” cannot be “mapped” in a way that is consistent with classical logic and probability theory.