Knot invariants and indefinite causal order

TL;DR

This paper investigates the concept of indefinite causal order in quantum spacetimes, introducing new measures, topological invariants, and knot-theoretic representations to understand and classify quantum causal structures.

Contribution

It introduces novel quantifiers and topological invariants for indefinite causal order, linking quantum causality with knot theory and providing an operational framework.

Findings

Definiteness of causal order is topologically invariant.

New measures quantify the degree of causal indefiniteness.

Knot-theoretic representations connect quantum causality with topology.

Abstract

We explore indefinite causal order between events in the context of quasiclassical spacetimes in superposition. We introduce several new quantifiers to measure the degree of indefiniteness of the causal order for an arbitrary finite number of events and spacetime configurations in superposition. By constructing diagrammatic and knot-theoretic representations of the causal order between events, we find that the definiteness or maximal indefiniteness of the causal order is topologically invariant. This reveals an intriguing connection between the field of quantum causality and knot theory. Furthermore, we provide an operational encoding of indefinite causal order and discuss how to incorporate a measure of quantum coherence into our classification.