Conjugate Logic

TL;DR

This paper introduces a conjugate logic framework that models quantum and quantum-like systems by encoding measurement predictions, including transformations between propositions on conjugate degrees of freedom, revealing different underlying models.

Contribution

It proposes a novel conjugate logic that captures quantum behaviors and demonstrates how different transformation choices lead to models like stabilizer quantum mechanics and Spekkens' toy theory.

Findings

Transformations enable modeling of correlations between systems.

Different transformation properties yield distinct quantum or quantum-like models.

The logic provides insights into what makes systems behave quantum or classical.

Abstract

We propose a conjugate logic that can capture the behavior of quantum and quantum-like systems. The proposal is similar to the more generic concept of epistemic logic: it encodes knowledge or perhaps more correctly, predictions about outcomes of future observations on some systems. For a quantum system, these predictions are statements about future outcomes of measurements performed on specific degrees of freedom of the system. The proposed logic will include propositions and their relations including connectives, but importantly also transformations between propositions on conjugate degrees of freedom of the systems. A key point is the addition of a transformation that allows to convert propositions about single systems into propositions about correlations between systems. We will see that subtle choices of the properties of the transformations lead to drastically different underlying mathematical models; one choice gives stabilizer quantum mechanics, while another choice gives Spekkens' toy theory. This points to a crucial basic property of quantum and quantum-like systems that can be handled within the present conjugate logic by adjusting the mentioned choice. It also enables a discussion on what behaviors are properly quantum or only quantum-like, relating to that choice and how it manifests in the system under scrutiny.