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arxiv: 2509.10700 · v3 · pith:UMAPBI7Mnew · submitted 2025-09-12 · 🪐 quant-ph · cond-mat.stat-mech

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

E. A. Ramirez Trino , M. A. Rajabpour This is my paper

Pith reviewed 2026-05-25 07:30 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mech
keywords stabilizer Renyi entropyShannon-Renyi entropyGaussian statesfree fermionstransverse-field Ising chainXX chainconformal field theoryquantum critical chains
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The pith

For any nondegenerate Gaussian eigenstate of quadratic fermions, the stabilizer Renyi entropy exactly equals the Shannon-Renyi entropy of a number-conserving free-fermion eigenstate on a doubled system.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes an exact equivalence between two types of Renyi entropies in quadratic fermion systems. Stabilizer Renyi entropy of a nondegenerate Gaussian eigenstate matches the Shannon-Renyi entropy of a related free-fermion state on twice the sites, measured in the computational basis. This mapping turns the transverse-field Ising ground state into the XX-chain ground state of length 2L, which yields closed expressions for specific indices and conformal scaling forms for critical chains.

Core claim

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 chain, the ground-state stabilizer entropies map to the Shannon-Renyi entropies of the XX-chain ground state of length 2L. Closed expressions follow for indices alpha equal to 1/2, 2, and 4; conformal-field-theory scaling laws hold at arbitrary index for periodic and open boundaries, with a discontinuity at alpha=4.

What carries the argument

The exact stabilizer-Shannon Renyi equivalence that maps any nondegenerate Gaussian eigenstate to the computational-basis Shannon-Renyi entropy of a number-conserving free-fermion state on a doubled system.

If this is right

  • Stabilizer entropies of the transverse-field Ising ground state equal Shannon-Renyi entropies of the XX-chain ground state of length 2L.
  • Closed expressions exist for stabilizer entropy at alpha equals 1/2, 2, and 4 in a broad class of critical closed free-fermion systems.
  • Conformal-field-theory scaling laws govern the stabilizer entropy at arbitrary Renyi index for critical systems with both periodic and open boundaries.
  • The scaling form at alpha equals 4 exhibits a discontinuity for both open and periodic boundary conditions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The mapping may let researchers obtain non-stabilizerness measures for other critical free-fermion models from existing Shannon-entropy results without new stabilizer calculations.
  • The discontinuity at alpha equals 4 could be checked numerically in finite-size critical chains to see whether it signals a change in how Renyi indices probe magic or universality.
  • If similar doublings exist beyond one dimension, the equivalence might connect stabilizer diagnostics to Shannon entropies in two-dimensional free-fermion critical points.

Load-bearing premise

The states are nondegenerate Gaussian eigenstates of quadratic fermion Hamiltonians.

What would settle it

Compute the stabilizer Renyi entropy at a chosen index for one explicit nondegenerate Gaussian eigenstate of a quadratic Hamiltonian and compare it directly to the Shannon-Renyi entropy of the corresponding doubled number-conserving free-fermion state in the computational basis.

read the original 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 $\alpha=\frac{1}{2},2,4$ 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 $\alpha=4$, these scaling forms display a discontinuity for both open and periodic boundary conditions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper establishes an exact equivalence: for any nondegenerate Gaussian eigenstate of a quadratic fermion Hamiltonian, the stabilizer Rényi entropy equals the computational-basis Shannon-Rényi entropy of a number-conserving free-fermion eigenstate on a doubled system. Specializing to the transverse-field Ising (TFI) chain, the ground-state stabilizer entropies map to those of the XX-chain ground state of length 2L. Closed-form expressions are derived for α=1/2, 2, 4 using additional identities, and CFT scaling laws (including a discontinuity at α=4) are obtained for critical free-fermion chains under periodic and open boundaries.

Significance. If the stated mapping and derivations hold, the result supplies an exact, parameter-free bridge between stabilizer Rényi entropy and Shannon-Rényi entropy for an important class of Gaussian states. This permits direct transfer of free-fermion techniques (including known CFT data for the TFI/XX models) to compute non-stabilizerness measures, yields concrete closed expressions at selected α, and produces falsifiable scaling predictions for critical chains. The explicit nondegeneracy hypothesis and scope to quadratic Hamiltonians are clearly delimited.

minor comments (2)
  1. The abstract and introduction should explicitly restate the nondegeneracy assumption when the mapping is first introduced, to avoid any ambiguity for readers who consult only those sections.
  2. Figure captions for the scaling plots (presumably in §4 or §5) should indicate the precise system sizes and boundary conditions used for the numerical checks of the CFT forms.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work, the clear summary of the main results, and the recommendation for minor revision. No specific major comments or points requiring clarification were raised in the report.

Circularity Check

0 steps flagged

Derivation is self-contained; no circular reductions identified

full rationale

The central claim is an exact mathematical equivalence between the stabilizer Renyi entropy of nondegenerate Gaussian eigenstates of quadratic fermion Hamiltonians and the computational-basis Shannon-Renyi entropy of a number-conserving free-fermion state on a doubled chain. This is derived directly from the algebraic properties of Gaussian states and the quadratic Hamiltonian structure, with the nondegeneracy condition stated explicitly. Closed expressions for specific indices (α=1/2,2,4) and CFT scaling forms follow from the mapping plus additional identities proved in the paper, without any fitted parameters, self-definitional loops, or load-bearing self-citations that reduce the result to its own inputs. All steps remain within the declared scope of free-fermion systems and are externally falsifiable via direct computation on the mapped states.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review based on abstract only; no explicit free parameters, invented entities, or ad-hoc axioms listed. The mapping rests on domain assumptions about Gaussian states.

axioms (1)
  • domain assumption States are nondegenerate Gaussian eigenstates of quadratic fermion Hamiltonians
    The equivalence is conditioned on this property of the eigenstates.

pith-pipeline@v0.9.0 · 5755 in / 1217 out tokens · 21847 ms · 2026-05-25T07:30:08.902244+00:00 · methodology

discussion (0)

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Reference graph

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