Yang-Baxter operators need quantum entanglement to distinguish knots

TL;DR

The paper demonstrates that non-entangling Yang-Baxter operators produce trivial knot invariants, establishing a fundamental link between quantum entanglement and topological knot invariants.

Contribution

It proves that quantum entanglement is necessary for Yang-Baxter operators to generate non-trivial knot invariants, connecting quantum information and topological properties.

Findings

Non-entangling Yang-Baxter operators yield trivial invariants

Quantum entanglement is essential for non-trivial knot invariants

Establishes a link between quantum entanglement and topological knot theory

Abstract

Any solution to the Yang-Baxter equation yields a family of representations of braid groups. Under certain conditions, identified by Turaev, the appropriately normalized trace of these representations yields a link invariant. Any Yang-Baxter solution can be interpreted as a two-qudit quantum gate. Here we show that if this gate is non-entangling, then the resulting invariant of knots is trivial. We thus obtain a general connection between topological entanglement and quantum entanglement, as suggested by Kauffman et al.