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Stabilizer Rényi Entropy and Conformal Field Theory

Masahiro Hoshino1,*, Masaki Oshikawa2,3, and Yuto Ashida1,4

  • *Contact author: hoshino-masahiro921@g.ecc.u-tokyo.ac.jp
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Phys. Rev. X 16, 011037 – Published 25 February, 2026

DOI: https://doi.org/10.1103/ylsz-dm3y

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 by 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.

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Quantum advantage depends on identifying resources that allow quantum machines to outperform classical computers. Nonstabilizerness—a measure of how far a state deviates from classically simulable stabilizer states—is one such essential resource. We establish a field-theoretical framework using boundary conformal field theory to reveal the universal long-distance behavior of nonstabilizerness in critical many-body systems. We find that the stabilizer Renyi entropy (SRE) of an entire system contains a universal size-independent term, while the mutual SRE exhibits logarithmic scaling governed by a universal coefficient. These analytical results, validated by numerical simulations of the Ising criticality, demonstrate that nonstabilizerness follows universal scaling laws analogous to entanglement in critical states. Beyond deepening our theoretical understanding, our results provide a diagnostic tool for characterizing quantum phases and establishing benchmarks for experiments in programmable quantum platforms.

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Phys. Rev. X 16, 011037– Published 25 February, 2026
  • Received 17 April 2025
  • Revised 25 September 2025
  • Accepted 22 January 2026
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