Quantum Covers in Quantum Measure Theory

In standard measure theory the measure on the total space Ω is normalised to one, which encodes the statement that “Ω happens”. The standard rules of measure theory tell us that the measure on Ω is strictly positive if and only if it cannot be covered by a collection of sets of zero measure. On the other hand, in quantum measure theory, simple examples suffice to demonstrate that a cover of sets of zero measure does not imply that the measure of Ω is zero. In this work we propose an appropriate generalisation of a cover to quantum measure theory, the quantum cover. A quantum cover is a collection of subsets of Ω such that, in addition to being a cover, it satisfies the property that if every one of its elements has zero quantum measure, then so does Ω. We show that any k-level inextendible antichain in the associated powerset lattice is a quantum cover, for Ω of finite cardinality. In addition, we show explicitly that a large class of “mixed level” inextendible antichains in this lattice are also quantum covers of Ω, when the quantum measure is obtained from a strongly positive decoherence functional, thus suggesting a universal characterisation. This construction is motivated in part by the recently proposed preclusionbased anhomomorphic logic approach to quantum interpretation.