Quantum networks theory

TL;DR

This paper extends quantum theory to network configurations, allowing superpositions of connections and flexible system partitioning, while maintaining core quantum properties through new concepts of consistency and comprehension.

Contribution

It introduces a novel mathematical framework for quantum networks with superposed connections and arbitrary logical partitions, preserving key quantum relations.

Findings

Quantum evolutions over network configurations are formalized.

Systems can be partitioned by arbitrary logical predicates.

Core quantum properties are maintained in the new framework.

Abstract

The formalism of quantum theory over discrete systems is extended in two significant ways. First, quantum evolutions are generalized to act over entire network configurations, so that nodes may find themselves in a quantum superposition of being connected or not, and be allowed to merge, split and reconnect coherently in a superposition. Second, tensors and traceouts are generalized, so that systems can be partitioned according to almost arbitrary logical predicates in a robust manner. The hereby presented mathematical framework is anchored on solid grounds through numerous lemmas. Indeed, one might have feared that the familiar interrelations between the notions of unitarity, complete positivity, trace-preservation, non-signalling causality, locality and localizability that are standard in quantum theory be jeopardized as the neighbourhood and partitioning between systems become both quantum, dynamical, and logical. Such interrelations in fact carry through, albeit two new notions become instrumental: consistency and comprehension.