Almost quantum theory

TL;DR

This paper reviews modal quantum theory (MQT), extends it to include mixed states and dynamics, and demonstrates its consistency with no-signalling principles despite lacking traditional quantum structures.

Contribution

It introduces extensions to MQT with mixed states and dynamics, and establishes a representation theorem analogous to quantum CP maps within a modal framework.

Findings

MQT can be extended to include mixed states and open system dynamics.

Possibility assignments in MQT are compatible with no-signalling.

A representation theorem for MQT maps is established.

Abstract

Modal quantum theory (MQT) is a "toy model" of quantum theory in which amplitudes are elements of a general field. The theory predicts, not the probabilities of a measurement result, but only whether or not a result is possible. In this paper we review MQT and extend it to include mixed states, generalized measurements and open system dynamics. Even though MQT does not have density operators, superoperators or any concept of "positivity", we can nevertheless establish a precise analogue to the usual representation theorem for CP maps. We also embed MQT in a larger class of modal theories. We show that the possibility assignments for separate measurements on a bipartite system in MQT are always weakly consistent with some probability assignment that respects the no-signalling principle.