Indefinite causal structures for continuous-variable systems

TL;DR

This paper extends the process matrix framework to continuous-variable quantum systems, addressing singularities and demonstrating interference effects in indefinite causal structures.

Contribution

It develops a continuous-variable generalization of the process matrix framework, solving singularity issues and providing an example of interference in indefinite causal orders.

Findings

Successfully generalizes the process matrix framework to infinite dimensions.

Identifies and resolves singularities in the continuous-variable extension.

Demonstrates interference effects in causal structures with continuous variables.

Abstract

In all our well-established theories, it is assumed that events are embedded in a global causal structure such that, for every pair of events, the causal order between them is always fixed. However, the possible interplay between quantum mechanics and general relativity may require a revision of this paradigm. The process matrix framework keeps the validity of quantum physics locally but does not assume the existence of a global causal order. It allows to describe causal structures corresponding to a quantum superposition of 'A is before B' and 'B is before A'. So far, the framework has been developed only for finite-dimensional systems, and a straightforward generalization to infinite dimensions leads to singularities. Such generalization is necessary for continuous-variable systems and is a prerequisite for quantum fields on indefinite causal structures. Here we develop the process-matrix framework for continuous-variable systems. We encounter and solve the problems of singularities. Moreover, we study an example of a process in infinite dimensions, and we derive correlations exhibiting intereference due to processes in which A is before B and B is before A.