Correlations and quantum circuits with dynamical causal order

TL;DR

This paper explores the concept of dynamical causal order in correlations and quantum circuits, introducing a new class of non-influenceable causal order and analyzing quantum processes with indefinite and dynamical causal structures.

Contribution

It identifies a new class of non-influenceable dynamical causal order and formalizes the relationship between indefinite and dynamical causal orders in quantum processes.

Findings

Discovered a new class of non-influenceable causal order.

Formalized the relationship between indefinite and dynamical causal orders.

Analyzed quantum circuits with classical and quantum control of causal order.

Abstract

Requiring that the causal structure between different parties is well-defined imposes constraints on the correlations they can establish, which define so-called causal correlations. Some of these are known to have a "dynamical" causal order in the sense that their causal structure is not fixed a priori but is instead established on the fly, with for instance the causal order between future parties depending on some choice of action of parties in the past. Here we identify a new way that the causal order between the parties can be dynamical: with at least four parties, there can be some dynamical order which can nevertheless not be influenced by the actions of past parties. This leads us to introduce an intermediate class of correlations with what we call non-influenceable causal order, in between the set of correlations with static (non-dynamical) causal order and the set of general causal correlations. We then define analogous classes of quantum processes, considering recently introduced classes of quantum circuits with classical or quantum control of causal order -- the latter being the largest class within the process matrix formalism known to have a clear interpretation in terms of coherent superpositions of causal orders. This allows us to formalise precisely in which sense certain quantum processes can have both indefinite and dynamical causal order.