Indefinite causal order in boxworld theories

TL;DR

This paper explores indefinite causal order within boxworld theories, constructing a higher-order theory with gbits, and demonstrates how physical principles can limit correlations, revealing stronger causal inequality violations than in quantum theory.

Contribution

It introduces a higher-order boxworld theory with gbits, proposes principles to constrain correlations, and compares causal inequality violations to quantum higher-order theories.

Findings

Full set of two-way signaling correlations recovered

Physical principles limit achievable correlations

Higher-order boxworld violates causal inequalities more than quantum theory

Abstract

An astonishing feature of higher-order quantum theory is that it can accommodate indefinite causal order. In the simplest bipartite setting, there exist signaling correlations for which it is fundamentally impossible to ascribe a definite causal order for the parties' actions. Moreover, the assumptions required to arrive at such a statement (local quantum transformations and well-behaved probabilities) result in a nontrivial set of correlations, whose boundary is, to date, uncharacterized. In this work, we investigate indefinite causal order in boxworld theories. We construct a higher-order theory whose descriptor is the generalized bit (gbit) - the natural successor of the classical bit and quantum qubit. By fixing the local transformations in boxworld and asking about the global causal structure, we find that we trivially recover the full set of two-way signaling correlations. In light of this, we motivate and propose two physical principles in order to limit the set of achievable correlations: nonsignaling preservation and no signaling without system exchange. We find that a higher-order boxworld theory that respects these physical principles leads to (i) a nontrivial set of achievable correlations and (ii) a violation of some causal inequalities that is higher than what can be achieved in higher-order quantum theory. These results lead us to conjecture that the set of correlations of our higher-order boxworld theory is an outer approximation to the set of correlations produced by higher-order quantum theory.