Picturing Indefinite Causal Structure

TL;DR

This paper introduces a framework for second-order causal processes that can handle indefinite causal structures, providing a diagrammatic proof that causality preservation extends to processes with ancillary systems.

Contribution

It characterizes the most general maps preserving causality, enabling analysis of indefinite causal structures and proving causality preservation in complex process scenarios.

Findings

Causality-preserving maps are characterized for second-order processes.

Causality preservation on separable processes implies preservation on processes with ancillas.

Preserving causality for separable processes is equivalent to doing so for strongly non-signalling processes.

Abstract

Following on from the notion of (first-order) causality, which generalises the notion of being tracepreserving from CP-maps to abstract processes, we give a characterization for the most general kind of map which sends causal processes to causal processes. These new, second-order causal processes enable us to treat the input processes as 'local laboratories' whose causal ordering needs not be fixed in advance. Using this characterization, we give a fully-diagrammatic proof of a non-trivial theorem: namely that being causality-preserving on separable processes implies being 'completely' causality preserving. That is, causality is preserved even when the 'local laboratories' are allowed to have ancilla systems. An immediate consequence is that preserving causality is separable processes is equivalence to preserving causality for strongly non-signalling (a.k.a. localizable) processes.