Connectomes and Properties of Quantum Entanglement

TL;DR

This paper explores how topological quantum field theories can model quantum entanglement through simple graph-based states, revealing properties like monogamy and enabling topological versions of quantum protocols.

Contribution

It introduces a topological framework for understanding entanglement, classifies entanglement types via connectome states, and constructs topological models of quantum communication protocols.

Findings

Classifies entanglement using graph-based connectome states.

Reveals topological properties of entanglement such as monogamy.

Constructs topological models of dense coding and teleportation.

Abstract

Topological quantum field theories (TQFT) encode properties of quantum states in the topological features of abstract manifolds. One can use the topological avatars of quantum states to develop intuition about different concepts and phenomena of quantum mechanics. In this paper we focus on the class of simplest topologies provided by a specific TQFT and investigate what the corresponding states teach us about entanglement. These ``planar connectome" states are defined by graphs of simplest topology for a given adjacency matrix. In the case of bipartite systems the connectomes classify different types of entanglement matching the classification of stochastic local operations and classical communication (SLOCC). The topological realization makes explicit the nature of entanglement as a resource and makes apparent a number of its properties, including monogamy and characteristic inequalities for the entanglement entropy. It also provides tools and hints to engineer new measures of entanglement and other applications. Here the approach is used to construct purely topological versions of the dense coding and quantum teleportation protocols, giving diagrammatic interpretation of the role of entanglement in quantum computation and communication. Finally, the topological concepts of entanglement and quantum teleportation are employed in a simple model of information retrieval from a causally disconnected region, similar to the interior of an evaporating black hole.