Consistent circuits for indefinite causal order

TL;DR

This paper introduces an extended circuit formalism using directed graphs and Boolean matrices to systematically construct and verify quantum processes with cyclic causal structures, ensuring their logical consistency.

Contribution

It provides a general method to build and analyze exotic quantum causal processes, including those violating causal inequalities, with an intuitive interpretative framework.

Findings

Framework defines validity rules for causal graphs.

Constructs consistent quantum processes from valid graphs.

Includes standard exotic processes within the generated class.

Abstract

Over the past decade, a number of quantum processes have been proposed which are logically consistent, yet feature a cyclic causal structure. However, there is no general formal method to construct a process with an exotic causal structure in a way that ensures, and makes clear why, it is consistent. Here we provide such a method, given by an extended circuit formalism. This only requires directed graphs endowed with Boolean matrices, which encode basic constraints on operations. Our framework (a) defines a set of elementary rules for checking the validity of any such graph, (b) provides a way of constructing consistent processes as a circuit from valid graphs, and (c) yields an intuitive interpretation of the causal relations within a process and an explanation of why they do not lead to inconsistencies. We display how several standard examples of exotic processes, including ones that violate causal inequalities, are among the class of processes that can be generated in this way; we conjecture that this class in fact includes all unitarily extendible processes.