Entropic limitations on fixed causal order

TL;DR

This paper introduces information inequalities that can certify quantum processes with indefinite causal order, demonstrated through violations by the quantum switch, advancing the understanding of quantum causality beyond fixed sequences.

Contribution

It develops a broad class of information inequalities valid for many measures, providing new tools to identify indefinite causal order in quantum processes.

Findings

Violations of inequalities by the quantum switch demonstrate indefinite causal order.

The inequalities hold for various information measures, including von Neumann and R'enyi entropies.

Strong subadditivity of quantum entropy underpins the method for detecting causal structures.

Abstract

Quantum processes can exhibit scenarios beyond a fixed order of events. We propose information inequalities that, when violated, constitute sufficient conditions to certify quantum processes without a fixed causal order -- causally separable or indefinite causal ordered processes. The inequalities hold valid for a vast class of information measures. Nevertheless, we take under scrutiny the von Neumann, $\alpha-$R\'enyi entropies with parameter $\alpha \in [1/2,1) \cup (1,\infty)$, and max- and min-entropies. We also discuss how the strong subadditivity of quantum (von Neumann) entropy, used along with the information inequality developed here, implies relevant witnesses of causally separable and indefinite causal ordered processes in marginal scenarios. Importantly, we show the violation of these inequalities for the quantum switch, a paradigmatic example of a process with indefinite causal order. Our approach contributes to the important research direction of information-theoretic characterization of quantum processes beyond fixed causal orders.