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

M. A. Rajabpour

TL;DR

The paper establishes an exact correspondence between stabilizer Rényi entropies and Shannon–Rényi entropies for Gaussian fermionic states, showing that the stabilizer entropy $M_{\alpha}(\rho)$ equals the Shannon–Rényi entropy $H_{\alpha}$ of a number-conserving free-fermion eigenstate on a doubled system in the computational basis. Specializing to the transverse-field Ising chain, the ground-state SRE at size $L$ maps to the SR entropies of the XX-chain ground state of length $2L$, enabling closed forms for $\alpha \in \{\tfrac{1}{2},2,4\}$ in a broad class of critical free-fermion systems via universal TFI functions. The authors derive conformal-field-theory scaling laws for SRE under periodic and open boundaries at arbitrary $\alpha$, unifying SR and SRE within the Gaussian framework and providing experimentally accessible avenues through local Pauli-basis measurements. The block-reduction framework reduces SRE computations to a single XX-chain kernel, with extensions to higher dimensions and non-critical regimes, broadening the universality and practical reach of these results.

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.

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

TL;DR

The paper establishes an exact correspondence between stabilizer Rényi entropies and Shannon–Rényi entropies for Gaussian fermionic states, showing that the stabilizer entropy equals the Shannon–Rényi entropy of a number-conserving free-fermion eigenstate on a doubled system in the computational basis. Specializing to the transverse-field Ising chain, the ground-state SRE at size maps to the SR entropies of the XX-chain ground state of length , enabling closed forms for in a broad class of critical free-fermion systems via universal TFI functions. The authors derive conformal-field-theory scaling laws for SRE under periodic and open boundaries at arbitrary , unifying SR and SRE within the Gaussian framework and providing experimentally accessible avenues through local Pauli-basis measurements. The block-reduction framework reduces SRE computations to a single XX-chain kernel, with extensions to higher dimensions and non-critical regimes, broadening the universality and practical reach of these results.

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 . Building on this correspondence, together with other exact identities we prove, closed expressions for the stabilizer entropy at indices 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.

Paper Structure

This paper contains 3 sections, 2 theorems, 49 equations, 2 tables.

Table of Contents

  1. Proof of Absolute-value Pfaffian–Minor Correspondence
  2. Block structure for $f(z)=z^{n}+1$.
  3. Block reduction for $f(z)=z^m+z^{-m}$ on the half--shifted grid

Key Result

Theorem 1

Assume $L=2M$ and $\bold{R}\in\mathbb{C}^{L\times L}$ be skew-symmetric and checkerboard up to permutation, i.e. there exists a permutation matrix $\bold{P}$ such that Then, for every real $\alpha> 0$, Note that, $\bold{Det}_{\alpha}(\bold{G})$ is invariant under the gauge $\bold{G}\mapsto \bold{D}_1\bold{\Pi}_1 \bold{G}\bold{\Pi}_2 \bold{D}_2$ for permutation matrices $\bold{\Pi}_{1,2}$ and dia

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2: (Polyphase/chiral block reduction to $f(z)=z+1$).