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arXiv:2509.10700 · Physical Review Letters

Equivalence of Stabilizer and Shannon Reńyi Entropies: Exact Results for Quantum Critical Chains

E. A. Ramirez Trino, M. A. Rajabpour

Published Aug 3, 2026• 1 citation
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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 α=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 α=4, these scaling forms display a discontinuity for both open and periodic boundary conditions.

DOI
10.1103/2frt-tdg9

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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 α=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 α=4, these scaling forms display a discontinuity for both open and periodic boundary conditions.
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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.
Local-basis Shannon entropies of critical wave functions contain universal subleading information, but the Shannon point of the critical transverse-field Ising chain is singular in the conventional Rényi approach. We treat it directly at n=1n=1n=1 by representing the complete computational-basis Born distribution as the odd-degree boundary of independent long-range Bernoulli edges. The Shannon chain rule separates the entropy into explicit independent-edge and conditional cycle-space terms. For the periodic chain this isolates the analytic edge anomaly and organizes the remaining constant by linked vertex support. Conditional on the stated finite-part matching and forest assumptions, every nonvanishing linked coefficient is represented by an explicit finite-dimensional integral, giving an all-orders analytic hierarchy; the four-vertex term is evaluated in closed form, and the first few linked sectors already nearly saturate the established periodic constant, which is also reconstructed independently from the exact finite-size distribution. For an interval, the exterior reduces exactly to one ghost vertex and the explicit squared logarithms cancel, leaving the analytic edge contribution γE=log⁡2/4−1/16γ_E=\log2/4-1/16γE​=log2/4−1/16. The conditional cycle term has an exact all-support decomposition into physical-line and ghost-linked sectors; conditional on closure of the boundary forest subtraction, these sectors define an all-orders hierarchy of renormalized line and monomer--dimer boundary periods for the remaining logarithmic coefficient. The first complete augmented linked coefficient is evaluated in closed form, with an exact cancellation between its line and ghost boundary-layer anomalies; an independent reconstruction from exact finite-size probabilities recovers the established coefficient γ1=0.060020(3)γ_1=0.060020(3)γ1​=0.060020(3).
We study the finite-temperature stabilizer Rényi entropy of the open critical quantum spin chains. At Rényi index one half, this observable probes the distribution of thermal Pauli-string expectation values and can be written as a sum over absolute values of all square minors of a finite-temperature correlation matrix for the transverse-field Ising chain. We show that this exponentially large sum is exactly reducible to a single Pfaffian. The Pfaffian representation reveals a block Toeplitz--Hankel structure and allows us to extract the large-size scaling in several thermal regimes. In the crossover window where the inverse temperature is proportional to the system size, the stabilizer entropy factorizes into a saturated extensive contribution and a universal finite-size scaling function. We find that this scaling function is a level-eight eta quotient, rather than the ordinary free-boundary Majorana thermal factor. The deviation is exponentially hidden at low temperature but controls the high-temperature crossover, where it gives a Cardy-like asymptotic for the Pauli-string expectation-weight spectrum. These results show that finite-temperature stabilizer entropy reveals hidden defect-like conformal data invisible to ordinary thermodynamic probes.
Magic, or nonstabilizerness, is the resource that promotes stabilizer operations to universal quantum computation. For bipartite pure states, its component intrinsic to the correlations between the subsystems, the nonlocal magic, is obtained by minimizing a magic measure over local unitaries. For the stabilizer Rényi entropy (SRE), the minimum is conjectured to be attained by the computational-basis (CB) representative, the state obtained by assigning the Schmidt coefficients in decreasing order to matching computational-basis labels. We prove this conjecture for two families of states at every system size and bipartition: states with dyadic-staircase Schmidt spectra and states of Schmidt rank at most six. For arbitrary bipartite pure states, we show that the CB value exceeds the nonlocal SRE by at most 444, fixing the leading term of any divergent scaling of nonlocal SRE. We further show that the nonlocal SRE grows at most logarithmically with the entanglement entropy. Consequently, one-dimensional area-law states have bounded nonlocal SRE, while critical states with logarithmic entanglement entropy permit at most doubly logarithmic growth with system size. We apply these results to the transverse-field Ising chain, where we tightly bound the nonlocal SRE in the gapped phases and identify its double-logarithmic growth at criticality.