The space of logically consistent classical processes without causal order

TL;DR

This paper geometrically models classical processes without predefined causal order, revealing they form a polytope with deterministic extremal points, and distinguishes natural from fine-tuned probabilistic processes.

Contribution

It introduces a geometric framework for classical causally unordered processes and characterizes a polytope of deterministic extremal points.

Findings

Processes form a polytope in the geometric model.

Deterministic processes are represented by extremal points.

Probabilistic processes requiring fine-tuning are excluded from the deterministic polytope.

Abstract

Classical correlations without predefined causal order arise from processes where parties manipulate random variables, and where the order of these interactions is not predefined. No assumption on the causal order of the parties is made, but the processes are restricted to be logically consistent under any choice of the parties' operations. It is known that for three parties or more, this set of processes is larger than the set of processes achievable in a predefined ordering of the parties. Here, we model all classical processes without predefined causal order geometrically and find that the set of such processes forms a polytope. Additionally, we model a smaller polytope --- the deterministic-extrema polytope --- where all extremal points represent deterministic processes. This polytope excludes probabilistic processes that must be --- quite unnaturally --- fine-tuned, because any variation of the weights in a decomposition into deterministic processes leads to a logical inconsistency.