Stabilizer Rényi Entropy Encodes Fusion Rules of Topological Defects and Boundaries

Abstract

We demonstrate that the stabilizer Rényi entropy (SRE), a computable measure of quantum magic, can serve as an information-theoretic probe for universal properties associated with conformal defects in one-dimensional quantum critical systems. Using boundary conformal field theory, we show that open boundaries manifest as a universal logarithmic correction to the SRE, whereas topological defects yield a universal size-independent term. When multiple defects are present, we find that the universal terms in the SRE faithfully reflect the defect-fusion rules that define a noninvertible symmetry algebra. These analytical predictions are corroborated by numerical calculations of the Ising model, where boundaries and topological defects are described by Cardy states and Verlinde lines, respectively.

Figures (5)

• Figure 1: (a) Conformal defects can be moved and fused by Clifford unitaries. The stabilizer Rényi entropy remains invariant under such operations, allowing it to directly probe the algebraic structure of defect fusion. (b) Open boundaries can be viewed as factorizing defects inserted in a periodic chain, whereas topological defects are created by locally altering the Hamiltonian.
• Figure 2: (a) When the system has open boundaries, the partition function $Z_{2\alpha}$ of the $2\alpha$-component theory in Eq. \ref{['eq:replica-trick']} is defined on the cylinder with two ends $A,B$ and a horizontal line defect due to the Bell projection at the fixed imaginary time $\tau\,{=}\,0$. After folding, the cylinder becomes a rectangle with four boundaries, and the number of fields is double. (b) Torus of size $L\,{\times}\,\beta$ with a topological defect $\mathcal{A}$ can be folded into a cylinder of circumference $L$ and length $\beta/2$ with a vertical line defect. Periodicity in the spatial direction is implicit in the top panel.
• Figure 3: (a) The SREs of open chains with the boundary pairs $(\mathcal{A}\,{\ast}\ket*{A},\ket*{B})$ and $(\ket*{A},\mathcal{A}\,{\ast}\ket*{B})$ are equivalent provided that one can move the defect $\mathcal{A}$ from one side to the other and fuse it with the boundary via a Clifford unitary. (b) The universal logarithmic correction is extracted by fitting the data $2M_2(L/2)-M_2(L)$ as a function of the system size $L$, where $M_2(L)$ is the $\alpha=2$ SRE of the $L$-qubit Ising critical state. The estimated coefficient agrees with the theoretical value $-1/4$ in both the $(\ket*{f},\ket*{f})$ and $(\ket*{\uparrow},\ket*{\uparrow})$ boundary pairs.
• Figure 4: Size-independent term $c_{2}^\mathcal{A}$ in the $\alpha\,{=}\,2$ SRE of the closed chain with (a) the identity defect $1$, (b) the $\mathbb{Z}_2$ defect $\eta$, and (c) the duality defect $\mathcal{D}$. The numerical data were obtained using the replica-Pauli MPS method tarabunga2024nonstabilizernessa. The data are extracted from the SRE fitted to $M_2=m_2L \,{-}\, c_2^{\mathcal{A}} + r/L$ with $L\in\qty{L_0{-}5,L_0{-}3,\ldots,L_0{+}5}$ and $L_0=11,13,\ldots,19$. The estimated universal values at the critical point $\lambda=1$ are $c_2^1=\ln\sqrt{2}$, $c_2^\eta=0.755(1)$, and $c_2^\mathcal{D}=0.020(3)$. (d) Size-independent term for two duality defects is extracted from the SRE of $L$ qubits fitted to $M_2=m_2(L-1) \,{-}\, c_2^{\mathcal{D}{\otimes}\mathcal{D}}$, accounting for the defect $\mathcal{T}^{-}$. The universal value reads $c_2^{\mathcal{D}\otimes\mathcal{D}} \,{=}\, \ln\sqrt{2}$.
• Figure S1: (Left) Finite-size scaling of the constant term $c_2$ extracted from the SRE of the Ising model with the candidate defect $H_{L1} = -Z_L - X_1 - X_L Z_1$. The lack of data collapse at the critical point signals that the defect is not topological. (Right) Extraction of the logarithmic term from the difference $2M_2(L/2) - M_2(L)$. The estimated coefficient $\approx -0.255$ is close to the value $-1/4$ characteristic of factorizing defects (open boundaries).