Entanglement Classification from a Topological Perspective

TL;DR

This paper introduces a topological approach to classifying quantum entanglement using TQFT, providing an intuitive framework that generalizes to multiple parties but with some limitations in capturing all entanglement types.

Contribution

It develops a topological classification scheme for entanglement based on TQFT, linking quantum states to topological equivalence classes and extending to multipartite systems.

Findings

Bipartite entanglement classification matches SLOCC under simple diagrams.

Multipartite classification captures some, but not all, SLOCC classes.

Connectome-based topological classification is intuitive and generalizable.

Abstract

Classification of entanglement is an important problem in Quantum Resource Theory. In this paper we discuss an embedding of this problem in the context of Topological Quantum Field Theories (TQFT). This approach allows classifying entanglement patterns in terms of topological equivalence classes. In the bipartite case a classification equivalent to the one by Stochastic Local Operations and Classical Communication (SLOCC) is constructed by restricting to a simple class of connectivity diagrams. Such diagrams characterize quantum states of TQFT up to braiding and tangling of the ``connectome.'' In the multipartite case the same restricted topological classification only captures a part of the SLOCC classes, in particular, it does not see the W entanglement of three qubits. Nonlocal braiding of connections may solve the problem, but no finite classification is attempted in this case. Despite incompleteness, the connectome classification has a straightforward generalization to any number and dimension of parties and has a very intuitive interpretation, which might be useful for understanding specific properties of entanglement and for design of new quantum resources.