Mapping embedded process matrices to spacetime games

Abstract

In 1935 Einstein first stated that quantum theory, as we know it, is an in- complete theory. In fact, quantum theory, as given by the Born rule, has an inherently random component, restricting it to providing probabilities for a cer- tain measurement outcome rather than specific predictions of outcomes. We explore a new approach that aims to lift this restriction with the help of game theory. It has been shown that Nashian game theory is incompatible with quantum theory. Hence, a new non-Nashian equilibrium has been introduced. This equilibrium is attained by relaxing the free choice assumption commonly found in quantum theory in favour of Einstein’s localism. The use of such a method is thought to enable making deterministic predictions over the out- comes of quantum experiments. There already exists a method for solving games with non-Nashian theory. It does, however, require as input a game in extensive form with imperfect infor- mation and therefore still misses the link to quantum experiments. We present a practical framework for representing quantum experiments mathematically based on the process matrix framework. This transition, from a physical experiment to the mathematical representation, makes use of the Choi-Jamiolkowski representation of quantum channels and states, as well as the process matrix representation of quantum processes. The implementation detailed in this report includes both a JSON storage for- mat and a Python library. This library comprises an algorithm for mapping a quantum experiment to a game in extensive form with imperfect information, as well as other (visualisation) utility functions. This report lays the way for future work to advance our understanding of non-Nashian game theory and to prove its consistency with quantum theory.