Analyzing reduced density matrices in SU(2) Chern-Simons theory

TL;DR

This work analyzes the reduced density matrices arising from torus-link complements $S^3\setminus T_{p,p}$ in $\mathrm{SU}(2)$ Chern-Simons theory at level $k$. By expressing the link-state via colored Jones invariants and performing a basis change with modular matrices, the authors obtain a diagonal eigenstructure for the reduced density matrix $Y_p$ with eigenvalues $\lambda_{p,a}=\frac{(\mathcal{S}_{0a})^{4-2p}}{N_p}$ and show that the characteristic polynomial $\mathrm{CP}_p(x)=\prod_{a=0}^k\left(x-\lambda_{p,a}\right)$ is monic of degree $k+1$ with rational coefficients. They derive explicit forms for $\mathrm{CP}_2$, $\mathrm{CP}_3$, $\mathrm{CP}_4$, and $\mathrm{CP}_5$ and provide a general construction for arbitrary $p$ in terms of $\mathrm{CP}_3$ and roots of unity, demonstrating a robust rational structure despite irrational eigenvalues. The results hint at underlying number-theoretic identities connecting entanglement spectra to modular data, and suggest natural extensions to other torus links $T_{p,q}$ with potential broader mathematical implications.

Abstract

We investigate the reduced density matrices obtained for the quantum states in the context of 3d Chern-Simons theory with gauge group SU(2) and Chern-Simons level $k$. We focus on the quantum states associated with the $T_{p,p}$ torus link complements, which is a $p$-party pure quantum state. The reduced density matrices are obtained by taking the $(1|p-1)$ bi-partition of the total system. We show that the characteristic polynomials of these reduced density matrices are monic polynomials with rational coefficients.