A diagrammatic language for the Causaloid framework

TL;DR

This paper introduces a diagrammatic language for the Causaloid framework, aiming to bridge the gap between indefinite causal structures and post-quantum theories by formalizing the framework's levels of physical compression.

Contribution

It provides a novel diagrammatic language and terminology for the Causaloid framework, facilitating its connection to Hilbert space-based approaches and post-quantum studies.

Findings

Developed a diagrammatic language for Causaloid framework

Defined new terminology for three levels of physical compression

Lays groundwork for linking Causaloid to Hilbert space approaches

Abstract

The Causaloid framework is an operational approach aimed to house both the radical aspects of General Relativity -- dynamic causal structure, and Quantum Theory -- indefiniteness, to provide a scaffolding that might be suitable for Quantum Gravity by providing a landscape of theories that allow for indefinite causal structure. One may consider it as a generalisation of generalised probability theories (or GPTs) where a priori regions are not assumed to have any given causal relationship, to incorporate the possibility of indefinite causal structure. Since its conception, there have been many advances in the field of indefinite causal structure mostly stemming from the work of Chiribella et al. on the quantum switch and supermaps and from Oreshkov et al. on causal inequalities and process matrices. These approaches have systems moving along wires and use Hilbert space structure. They violate the standard causality constraints of Quantum Theory and, in this sense, can be regarded as post-quantum. The Causaloid approach does not necessarily have systems moving along wires or Hilbert spaces. This is the first paper in a trilogy of papers aiming to close the gap between the Causaloid (that allows for GPTs) and post-quantum studies that employ Hilbert spaces. To do so in the present paper, we provide a diagrammatic language for the Causaloid framework along with new terminology for the three levels of physical compression (called Tomographic, Compositional, and Meta compression).