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.
Highlights
Context, the simple gates stabilize a set of generalized Pauli operators and the simple states prepared by their circuits are stabilizer states
We study the extent to which knot and link states can or cannot be described by stabilizer states
By implementing a particular magic monotone known as the “mana” we quantify the magic of knot and link states
Summary
Let us briefly review the construction of the states in question (a more detailed construction can be found in [25]). One can imagine that one such basis state, |j , is prepared by the path-integration over the solid torus with a Wilson line in the Rj representation threading the interior longitude. We will call this basis the representation basis (or rep basis) to distinguish it from a different choice of basis that will follow. Is given by the path-integral over the manifold resulting from gluing S3 \ N (L) to the solid tori preparing each |ji (with their orientation reversed) along their common boundary This has the effect of filling in each tubular neighborhood of N (L) along with a Wilson loop along each circle, Li, This is depicted as a in the conjugate cartoon in figure representation ji∗: Wji∗(Li) = 1.
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