Stabilizer-Shannon Renyi Equivalence: Exact Results for Quantum Critical Chains

Abstract

Shannon-Renyi and stabilizer entropies are key diagnostics of structure, non-stabilizerness, phase transitions, and universality in quantum many-body states. We establish an exact correspondence for quadratic fermions: for any nondegenerate Gaussian eigenstate, the stabilizer Renyi entropy equals the Shannon-Renyi 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 maps to the Shannon-Renyi entropies of the XX-chain ground state of length $2L$. Building on this correspondence, together with other exact identities we prove, closed expressions for the stabilizer entropy at indices $α=\frac{1}{2},2,4$ for a broad class of critical closed free-fermion systems were derived. Each of these can be written with respect to the universal functions of the TFI chain. We further obtain conformal-field-theory scaling laws for the stabilizer entropy under both periodic and open boundaries at arbitrary Renyi index for these critical systems. At $α=4$, these scaling forms display a discontinuity for both open and periodic boundary conditions.