No-signalling constrains quantum computation with indefinite causal structure

TL;DR

This paper develops a formal framework for quantum computation involving indefinite causal structures, characterizing the composition rules of higher order quantum maps constrained by no-signalling conditions.

Contribution

It introduces a formalism for quantum computation with indefinite causal order, linking composition rules to physical signalling constraints within higher order quantum maps.

Findings

Characterization of admissible compositions of higher order quantum maps.

Identification of rules determined by signalling relations.

Framework compatible with quantum theory's mathematical structure.

Abstract

Quantum processes with indefinite causal structure emerge when we wonder which are the most general evolutions, allowed by quantum theory, of a set of local systems which are not assumed to be in any particular causal order. These processes can be described within the framework of higher-order quantum theory which, starting from considering maps from quantum transformations to quantum transformations, recursively constructs a hierarchy of quantum maps of increasingly higher order. In this work, we develop a formalism for quantum computation with indefinite causal structures; namely, we characterize the computational structure of higher order quantum maps. Taking an axiomatic approach, the rules of this computation are identified as the most general compositions of higher order maps which are compatible with the mathematical structure of quantum theory. We provide a mathematical characterization of the admissible composition for arbitrary higher order quantum maps. We prove that these rules, which have a computational and information-theoretic nature, are determined by the more physical notion of the signalling relations between the quantum systems of the higher order quantum maps.