Stabilizer Rényi Entropy and Conformal Field Theory

TL;DR

This work establishes a field-theoretical BCFT framework to understand the universal aspects of nonstabilizerness, quantified by the Stabilizer Rényi Entropy (SRE), in 1+1D critical quantum states. By identifying the SRE with a participation entropy in the Bell basis of a doubled Hilbert space, the authors show that the universal full-state term is governed by the g-factor of a replicated boundary state, while the universal logarithmic scaling of the mutual SRE is dictated by the scaling dimension of a boundary condition changing operator. The Ising critical point is worked out in detail via bosonization to an S^1/ℤ2 orbifold CFT, yielding analytic results c_alpha = (ln√α)/(α−1) and Δ_{2α} = 1/16, which are corroborated numerically using tensor networks and perfect Pauli sampling. The results provide a principled link between nonstabilizerness and BCFT data, offering a predictive tool for universal features of quantum resources in critical many-body systems and a pathway for experimental benchmarking with Bell-state measurements on critical states.

Abstract

Understanding universal aspects of many-body systems is one of the central themes in modern physics. Recently, the stabilizer Rényi entropy (SRE) has emerged as a computationally tractable measure of nonstabilizerness, a crucial resource for fault-tolerant universal quantum computation. While numerical results suggested that the SRE in critical states can exhibit universal behavior, its comprehensive theoretical understanding has remained elusive. In this work, we develop a field-theoretical framework for the SRE in a $(1+1)$-dimensional many-body system and elucidate its universal aspects using boundary conformal field theory. We demonstrate that the SRE is equivalent to a participation entropy in the Bell basis of a doubled Hilbert space, which can be calculated from the partition function of a replicated field theory with the interlayer line defect created by the Bell-state measurements. This identification allows us to characterize the universal contributions to the SRE on the basis of the data of conformal boundary conditions imposed on the replicated theory. We find that the SRE of the entire system contains a universal size-independent term determined by the noninteger ground-state degeneracy known as the g-factor. In contrast, we show that the mutual SRE exhibits the logarithmic scaling with a universal coefficient given by the scaling dimension of a boundary condition changing operator, which elucidates the origin of universality previously observed in numerical results. As a concrete demonstration, we present a detailed analysis of the Ising criticality, where we analytically derive the universal quantities at arbitrary Rényi indices and numerically validate them with high accuracy by employing tensor network methods. These results establish a field-theoretical approach to understanding the universal features of nonstabilizerness in quantum many-body systems.

Figures (14)

• Figure 1: Summary of the main results. (a) Comparison of the long-distance behavior of entanglement and nonstabilizerness in critical states. The universal feature of entanglement manifests itself as the logarithmic scaling with the coefficient given by the central charge $c$ of the corresponding CFT. In contrast, we show that the universality of nonstabilizerness is encoded in the size-independent term $c_\alpha$ in the SRE $M_{\alpha}$ and the coefficient of the logarithmic scaling in the mutual SRE $W_\alpha$, which are given by the g-factor $g_1$ and the scaling dimension $\Delta_{2\alpha}$ of a boundary condition changing operator (BCCO) $\mathcal{B}_{2\alpha}(x)$, respectively. Here, the Rényi index is denoted by $\alpha$. (b) From a field-theoretical perspective, the SRE of the entire system is obtained by the ratio between the partition functions with and without the measurement-induced interlayer line defect in the $2\alpha$-fold replicated field theory. The universal term $c_\alpha$ in the SRE can then be determined by the g-factor $g_1$ of the corresponding conformal boundary state. (c) The long-distance behavior of the mutual SRE is governed by the partition function with two different boundary conditions (indicated by the red and green boundaries). This quantity can be expressed as a two-point correlation function $\langle\mathcal{B}_{2\alpha}(l)\mathcal{B}_{2\alpha}(0)\rangle$ of the corresponding BCCO. The resulting logarithmic scaling behavior is characterized by the universal coefficient determined by the scaling dimension $\Delta_{2\alpha}$ of $\mathcal{B}_{2\alpha}(x)$.
• Figure 2: Schematic figure illustrating how the Bell basis is defined in the doubled spin chain. The SRE is calculated from the Born probability of the Bell-state measurements across the two chains. The measurement outcomes are labeled by $\vec{m}\in\qty{0,1}^{2L}$.
• Figure 3: (a) The partition function $Z_{2\alpha}$ of the $2\alpha$-component theory is defined on the torus of size $L\times \beta$. The figure only illustrates periodicity in the imaginary-time direction. (b) The system is folded into the $4\alpha$-component theory defined on the cylinder of circumference $L$ and length $\beta/2$ with boundary conditions $\Gamma_1$ at $\tau=0$ and $\Gamma_2$ at $\tau=\beta/2$. If we write the original fields as $\varphi_i\,(i=1,2,\ldots,2\alpha)$, then the remaining $2\alpha$ fields $\varphi_j\,(j=2\alpha+1,2\alpha+2,\ldots,4\alpha)$ satisfy the sewing condition $\varphi_i=\varphi_{i+2\alpha}$ at the boundaries $\Gamma_1$ and $\Gamma_2$.
• Figure 4: The partition function $Z_{2\alpha}(A)$ is calculated from the path-integral on a cylinder of circumference $L$ and length $\beta/2$ with the boundary condition $\Gamma_2$ at $\tau=\beta/2$ and two different boundary conditions at $\tau=0$: $\Gamma_1$ in $0\leq x\leq l$ and $\Gamma_0$ in $l\leq x\leq L$. There are two boundary condition changing points at $x=0,l$, and the partition function $Z_{2\alpha}(A)$ can be calculated from a two-point correlation function of the BCCOs $\mathcal{B}_{2\alpha}(x)$ inserted at these points.
• Figure 5: Schematic illustration of the $\mathbb{Z}_2$ orbifold construction. Left: The $S^1$ free boson is defined on a circle with radius $R$. Right: The $S^1\!/\mathbb{Z}_2$ free-boson CFT is obtained by identifying points related by the $\mathbb{Z}_2$ transformation $\phi\to-\phi$, effectively folding the circle into the line segment $[0,\pi]$. The crosses indicate the fixed points ($\phi=0,\pi$) of the $\mathbb{Z}_2$ orbifold.