Dynamics of quantum causal structures

TL;DR

This paper develops a framework for the dynamics of quantum causal structures, showing that causal order is preserved under reversible transformations but can change under nonreversible ones, exemplified by the quantum switch.

Contribution

It introduces a formalism for the transformations of process matrices, expanding understanding of how quantum causal structures evolve and change.

Findings

Causal order remains invariant under continuous, reversible transformations.

Nonreversible transformations can alter the causal order of operations.

The quantum switch exemplifies causal order change via superposition.

Abstract

It was recently suggested that causal structures are both dynamical, because of general relativity, and indefinite, due to quantum theory. The process matrix formalism furnishes a framework for quantum mechanics on indefinite causal structures, where the order between operations of local laboratories is not definite (e.g. one cannot say whether operation in laboratory A occurs before or after operation in laboratory B). Here we develop a framework for "dynamics of causal structures", i.e. for transformations of process matrices into process matrices. We show that, under continuous and reversible transformations, the causal order between operations is always preserved. However, the causal order between a subset of operations can be changed under continuous yet nonreversible transformations. An explicit example is that of the quantum switch, where a party in the past affects the causal order of operations of future parties, leading to a transition from a channel from A to B, via superposition of causal orders, to a channel from B to A. We generalise our framework to construct a hierarchy of quantum maps based on transformations of process matrices and transformations thereof.