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Establish the logarithmic negativity and even-moment formulas for U(1)k three-component link states

Establish that for a three-component link state in U(1)k Chern–Simons theory with Gauss linking numbers L_AB, L_BC, and L_CA, the even moments of the partially transposed reduced density matrix satisfy Tr[(ρ_{BC}^{Γ})^{2p}] = 1 / [ k^{2p−1} · gcd(k,L_CA,L_AB,L_BC)^{2p−2} · gcd(k,L_CA,L_AB) ] for all integers p ≥ 1, and that the logarithmic negativity obeys E_N(B;C) = log[ gcd(k,L_CA,L_AB) / gcd(k,L_CA,L_AB,L_BC) ].

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Background

Using the replica trick, the authors compute Tr[(ρ{BC}{Γ}){2}] exactly and obtain an expression for Tr[(ρ{BC}{Γ}){2p}] when L_CA=0, then propose a general symmetric formula together with the resulting logarithmic negativity expression.

These formulas link entanglement negativity to arithmetic data (gcds of k and pairwise linking numbers), but are currently supported only by numerical checks; a complete analytic proof is lacking.

References

We note again that the expressions eq-tracenorm-2p and eq-negativity-3link are conjectures, motivated by the rigorous results eq-EN-replica2 and eq-EN-replica2p, and further supported by extensive numerical tests.

eq-tracenorm-2p:

Tr(ρBCΓ)2p=1k2p−1gcd⁡(k,LCA,LAB,LBC)2p−2gcd⁡(k,LCA,LAB) .\mathrm{Tr}\big(\rho_{BC}^{\Gamma}\big)^{2p} = \frac{1}{k^{2p - 1}}\gcd(k,L_{CA},L_{AB},L_{BC})^{2p - 2}\gcd(k,L_{CA},L_{AB})\ .

eq-negativity-3link:

EN(B;C)=lim⁡p→1/2log⁡Tr(ρBCΓ)2p=log⁡gcd⁡(k,LCA,LAB)gcd⁡(k,LCA,LAB,LBC) .E_{\mathcal{N}}(B;C) = \lim_{p\to 1/2} \log \mathrm{Tr}\big(\rho_{BC}^{\Gamma}\big)^{2p} = \log\frac{\gcd(k,L_{CA},L_{AB})}{\gcd(k,L_{CA},L_{AB},L_{BC})}\ .

eq-EN-replica2:

Tr(ρBCΓ)2= ⁣∑β1,2,χ1,2 ⁣1k4η ⁣[(β1 ⁣− ⁣β2)LAB−(χ1 ⁣− ⁣χ2)LCA] η ⁣[(β2 ⁣− ⁣β1)LAB−(χ2 ⁣− ⁣χ1)LCA]=1k4×k2×kgcd⁡(k,LCA,LAB)=1kgcd⁡(k,LCA,LAB) ,\begin{split} \mathrm{Tr}\big(\rho_{BC}^{\Gamma}\big)^{2} &=\! \sum_{\beta_{1,2}, \chi_{1,2}}\! \frac{1}{k^4} \eta\!\left[(\beta_1\!-\!\beta_2)L_{AB} - (\chi_1\!-\!\chi_2)L_{CA}\right]\, \eta\!\left[(\beta_2\!-\!\beta_1)L_{AB} - (\chi_2\!-\!\chi_1)L_{CA}\right]\\ &= \frac{1}{k^4}\times k^2 \times k\gcd(k,L_{CA},L_{AB})\\ &= \frac{1}{k} \gcd(k,L_{CA},L_{AB})\ , \end{split}

eq-EN-replica2p:

Tr(ρBCΓ)2p∣LCA=0=1k2p−1gcd⁡(k,LAB,LBC)2p−2gcd⁡(k,LAB) .\mathrm{Tr}\big(\rho_{BC}^{\Gamma}\big)^{2p}\Big|_{L_{CA} = 0} = \frac{1}{k^{2p - 1}}\gcd(k,L_{AB},L_{BC})^{2p - 2}\gcd(k,L_{AB})\ .

— Multi-entropy from Linking in Chern-Simons Theory (2510.18408 - Yuan et al., 21 Oct 2025) in Section 4 (Logarithmic negativity for three-component links), around Eqs. (4.10)–(4.12)