Equivalence of Stabilizer and Shannon R\'enyi Entropies: Exact Results for Quantum Critical Chains

Abstract:Shannon-Rényi and stabilizer entropies are key diagnostics of structure, ``nonstabilizerness,'' phase transitions, and universality in quantum many-body states. We establish an exact correspondence for quadratic fermions: for any Gaussian eigenstate, the stabilizer Rényi entropy equals the Shannon-Rényi entropy of a number-conserving free-fermion eigenstate on a doubled system, evaluated in the computational basis. Specializing to the transverse-field Ising (TFI) chain, the TFI ground-state stabilizer entropies map to the Shannon-Rényi entropies of the XX-chain ground state of length $2L$. Building on this correspondence, together with other exact identities we prove, we derive closed expressions for the stabilizer entropy at indices $\alpha=\tfrac{1}{2},2,4$ for a broad class of critical closed free-fermion systems. Each of these can be written with respect to the universal functions of the TFI chain. We further derive conformal-field-theory scaling laws for the stabilizer entropy at arbitrary Rényi index under both periodic and open boundary conditions. At $\alpha=4$, these scaling forms display a discontinuity for both open and periodic boundary conditions.

Quantum Physics (quant-ph); Statistical Mechanics (cond-mat.stat-mech)

arXiv:2509.10700 [quant-ph]

(or arXiv:2509.10700v4 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2509.10700 Focus to learn more arXiv-issued DOI via DataCite

Phys. Rev. Lett. 137, 090403 (2026)

https://doi.org/10.1103/2frt-tdg9 Focus to learn more DOI(s) linking to related resources

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