Achieving Maximal Causal Indefiniteness in a Maximally Nonlocal Theory

TL;DR

This paper explores superposition and indefinite causal order within a generalized probabilistic framework, revealing that composite systems can admit superposition without entanglement, and demonstrating a maximally nonlocal theory with post-quantum causal violations.

Contribution

It introduces a new notion of superposition in generalized theories, shows that composite systems can admit superposition independently of entanglement, and provides an example of a maximally nonlocal theory with indefinite causal order.

Findings

Single system state-spaces do not admit superposition in maximal theories.

Composite systems can admit superposition without entanglement.

A maximally Bell-nonlocal theory allows post-quantum violations of causal inequalities.

Abstract

Quantum theory allows for the superposition of causal orders between operations, i.e., for an indefinite causal order; an implication of the principle of quantum superposition. Since a higher theory might also admit this feature, an understanding of superposition and indefinite causal order in a generalised probabilistic framework is needed. We present a possible notion of superposition for such a framework and show that in maximal theories, respecting non-signalling relations, single system state-spaces do not admit superposition; however, composite systems do. Additionally, we show that superposition does not imply entanglement. Next, we provide a concrete example of a maximally Bell-nonlocal theory, which not only admits the presented notion of superposition, but also allows for post-quantum violations of theory-independent inequalities that certify indefinite causal order; even up to an algebraic bound. These findings might point towards potential connections between a theory's ability to admit indefinite causal order, Bell-nonlocal correlations and the structure of its state spaces.