Knots, links, and long-range magic

TL;DR

This paper investigates the quantum resource theory of knot and link states in 3d Chern-Simons theory, demonstrating that these states are generally highly magical, especially in their long-range correlations.

Contribution

It introduces a quantification of magic in knot and link states using the mana monotone and shows that these states are typically highly magical, with emphasis on long-range magic correlations.

Findings

Knot and link states are generally highly magical in SU(2)_k Chern-Simons theory.

Long-range correlations in link states contribute significantly to their magic.

Numerical results indicate most link states exhibit predominantly long-range magic.

Abstract

We study the extent to which knot and link states (that is, states in 3d Chern-Simons theory prepared by path integration on knot and link complements) can or cannot be described by stabilizer states. States which are not classical mixtures of stabilizer states are known as "magic states" and play a key role in quantum resource theory. By implementing a particular magic monotone known as the "mana" we quantify the magic of knot and link states. In particular, for $SU(2)_k$ Chern-Simons theory we show that knot and link states are generically magical. For link states, we further investigate the mana associated to correlations between separate boundaries which characterizes the state's long-range magic. Our numerical results suggest that the magic of a majority of link states is entirely long-range. We make these statements sharper for torus links.