Plausibility measures on test spaces

Abstract

Plausibility measures are structures for reasoning in the face of uncertainty that generalize probabilities, unifying them with weaker structures like possibility measures and comparative probability relations. So far, the theory of plausibility measures has only been developed for classical sample spaces. In this paper, we generalize the theory to test spaces, so that they can be applied to general operational theories, and to quantum theory in particular. Our main results are two theorems on when a plausibility measure agrees with a probability measure, i.e. when its comparative relations coincide with those of a probability measure. For strictly finite test spaces we obtain a precise analogue of the classical result that the Archimedean condition is necessary and sufficient for agreement between a plausibility and a probability measure. In the locally finite case, we prove a slightly weaker result that the Archimedean condition implies almost agreement.