October 2024

The conjecture of causal decompositions proposes that there is a deep link between two fundamental structures of (unitary) quantum theory: causal structure and compositional structure. It states that, for any unitary U from several inputs to several outputs, U features a certain causal structure (i.e. satisfies a certain set of no-influence relations between inputs and outputs) if and only if it can be decomposed into the composition of ‘smaller’ unitaries, along a graph that makes the causal structure evident. This would provide a very important equivalence between an empirical, operationally accessible notion (causal structure) and a powerful mathematical structure (compositional structure). This conjecture has so far been proven in cases involving a few inputs and outputs. Here, I will present a result (to appear soon) showing that causal decompositions also exist for 1D Quantum Cellular Automata (QCAs). 1D QCAs can be seen as representing lightcone-abiding dynamics in a discretised (1+1)D Minkowski spacetime. This result can be understood as showing that, in one (discretised) spatial dimension, a dynamics satisfies relativistic causality if and only if it can be broken down into a series of nearest-neighbour interactions. It also provides the first constructive form of 1D QCAs at any causality radius. On the way, I will also discuss how the proof of this result involves important conceptual and mathematical innovations, related to the use of C* algebras to model quantum subsystems, which might find application elsewhere as well.