Composing decoherence functionals

TL;DR

This paper investigates the composability issues of decoherence functionals in Quantum Measure Theory, revealing fundamental limitations on their combination and the exclusive role of strongly positive DFs in maintaining consistency.

Contribution

It demonstrates the deep limitations of composability in QMT and proves that only strongly positive DFs can form closed, composable sets consistent with quantum DFs.

Findings

Composability of DFs is fundamentally limited, with some DFs incompatible with multiple copies.

The set of strongly positive DFs cannot be expanded while remaining closed under composition.

Any closed set of DFs containing all quantum DFs must consist solely of strongly positive DFs.

Abstract

Quantum Measure Theory (QMT) is a generalization of quantum theory where physical predictions are computed from a matrix known as \emph{decoherence functional} (DF). Previous works have noted that, in its original formulation, QMT exhibits a problem with composability, since the composition of two decoherence functionals is, in general, not a valid decoherence functional. This does not occur when the DFs in question happen to be positive semidefinite (a condition known as strong positivity). In this paper, we study the concept of composability of DFs and its consequences for QMT. Firstly, we show that the problem of composability is much deeper than originally envisaged, since, for any $n$, there exists a DF that can co-exist with $n-1$ copies of itself, but not with $n$. Secondly, we prove that the set of strongly positive DFs cannot be enlarged while remaining closed under composition. Furthermore, any closed set of DFs containing all quantum DFs can only contain strongly positive DFs.