Causal structure in the presence of sectorial constraints, with application to the quantum switch

Abstract

Existing work on quantum causal structure assumes that one can perform arbitrary operations on the systems of interest. But this condition is often not met. Here, we extend the framework for quantum causal modelling to cases where a system can suffer sectorial contraints, that is, restrictions on the orthogonal subspaces of its Hilbert space that may be mapped to one another. Our framework (a) proves that a number of different intuitions about causal relations turn out to be equivalent; (b) shows that quantum causal structures in the presence of sectorial constraints can be represented with a directed graph; and (c) defines a fine-graining of the causal structure in which the individual sectors of a system bear causal relations, which provides a more detailed analysis than its coarse-grained counterpart. As an example, we apply our framework to purported photonic implementations of the quantum switch to show that while their coarse-grained causal structure is cyclic, their fine-grained causal structure is acyclic. We therefore conclude that these experiments realize indefinite causal order only in a weak sense. Notably, this is the first argument to this effect that is not rooted in the assumption that the causal relata must be localized in spacetime.

Which theories have a measurement problem?

“…Applying the interchange law (14) to pull E 1 2 up, and then ( 31), (29), and (28), we infer the following.…”

Operational interpretation of the vacuum and process matrices for identical particles

“…Let us now consider the optical quantum switch, a similar protocol in which the notion of space time the vacuum also plays a role. Current optical implementations of the quantum switch feature four spacetime points, the same as the dSD protocol [9,10,11,12], thus having the similar type of the spacetime schematic description, see Figure 2. However, by introducing the notion of time delocalised operations it was argued that the optical switch implements only two operations, U in spacetime points A or A , and V in spacetime points B or B [21].…”

“…One of the interesting aspects of the quantum switch is that it gives rise to a superposition of orders of quantum operations. In a recent work [9], the difference between the superposition of orders of quantum operations and the superposition of causal orders in spacetime was discussed in detail, and it was demonstrated that the latter can in principle be realised only in the context of quantum gravity (see also [10,11,12]). The detailed analysis of the causal structure of the quantum switch has revealed one important qualitative aspect of the process matrix description -in order to properly account for the causal structure of an arbitrary process, it is necessary to introduce the notion of the quantum vacuum as a possible physical state.…”

Consistent circuits for indefinite causal order

“…out to M β in , via the characterisation of causal influence in terms of algebras [33] (note that it makes sense to talk about the factors M β out and M β in of its input and output spaces because the ζ(M β )'s do not act on these). Therefore, one can find a unitary causal decomposition of it as…”