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Stabilizer R\'enyi moments beyond entanglement

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Vishnu Jejjala, Joydeep Naskar

Quantum Physics · Submitted 20 Oct 2026 · 51 references

Pith's verdict: Solid exact results for stabilizer moments at fixed support, but the continuum CFT exponent relies on unproven assumptions; still worth engaging.

T0 · AI review

How this review was made: An AI referee (deepseek/deepseek-v4-flash) read the paper on October 21, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

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arXiv ID2610.27596 (version 1)
Pith IDZ4TL4T3X, Pith's permanent record of this paper and its review
CitationJoydeep Naskar, Vishnu Jejjala. "Stabilizer R\'enyi moments beyond entanglement." arXiv:2610.27596 (2026). https://pith.science/paper/Z4TL4T3X
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The paper

The paper claims

An on-site rotation changes the decay exponent of stabilizer moments while preserving all entanglement spectra.

The reading

Stabilizer Rényi moments measure quantum magic, but in mixed states they also pick up entanglement contributions. This paper shows that for reductions of pure stabilizer states, the moment equals the Rényi-2 entropy to the power α−1, providing a baseline that must be subtracted. The authors derive a symmetry selection rule: in a state invariant under an on-site symmetry, only oppositely charged Weyl operators can correlate across separated regions, fixing the leading decay exponent at fixed support to 2α times the smallest charged scaling dimension. An exact Ising example demonstrates that an on-site rotation changes the full decay exponent from min{α/4,1} to 1/4 for integer α≥2, while preserving every subsystem entanglement spectrum—proving that the operator basis can affect the leading exponent, not just the amplitude. For growing Ising intervals at index two, a Pfaffian representation yields the charged amplitude 1/4, consistent with a candidate conformal boundary calculation, though endpoint matching and control of the order of limits remain open.

What carries the argument

The symmetry selection rule (1.2) shows that in a symmetry-invariant state, charged Weyl expectations vanish on single regions, so separated-region moments only receive contributions from oppositely charged pairs, with power set by the smallest charged scaling dimension. For the Ising model, the exact Pfaffian representation (5.7) of the index-two charged coefficient uses a confluent Vandermonde identity to convert sums over Pauli strings into Pfaffians of skew-symmetric moment matrices, enabling evaluation up to 128 sites. The candidate conformal boundary calculation (Appendix B.2) models the measurement projector as an orbifold Dirichlet-Neumann junction and predicts a charged coupling of 1/2, leading to the amplitude 1/4.

Extended reading notes

Core claim

The paper establishes that stabilizer Rényi moments of reduced density matrices are not determined by entanglement spectra alone. For intervals in critical quantum chains described by two-dimensional conformal field theories, the authors prove an exact entanglement baseline (cM_α(A) = S_2(A)/(α−1) for reductions of pure stabilizer states) and a symmetry selection rule (the leading charged contribution at fixed support decays as $η^{{2αΔ^{ch}}$_{min}} where η is the cross-ratio). The central explicit example uses the critical Ising chain: an on-site non-Clifford rotation changes the full decay exponent of the two-site ratio from min{α/4,1} to 1/4 for α≥2 while preserving every subsystem entanglement spectrum, showing that the operator completion affects the leading spatial scaling. At index two, the authors obtain a Pfaffian formula for the charged coefficient and evaluate it up to 128 sites; the data support a limiting value of 1/4, consistent with the orbifold boundary calculation, but a rigorous proof requires uniform remainder bounds and endpoint identification.

Load-bearing premise

The identification of the continuum exponent relies on unproven uniform remainder bounds (4.11) and assumptions (A1)–(A4) concerning the conformal limit of the measured boundary, the vanishing of the single-sheet neutral coefficient, and the absence of lower neutral powers.

Editorial extensions

If this is right

  • Stabilizer Rényi moments can distinguish operator-basis choices that entanglement cannot, providing a finer probe of quantum correlations in critical systems.
  • The decay of stabilizer moments at fixed support encodes the scaling dimensions of charged operators under the on-site symmetry, offering a lattice route to extract conformal data.
  • The entanglement baseline must be subtracted from mixed-state stabilizer moments to isolate magic, but the residual is not a faithful measure of mixed-state magic, only a measure of non-flatness of the Weyl participation distribution.
  • The Pfaffian method at index two provides an efficient way to compute stabilizer moments in free-fermion chains for intervals up to hundreds of sites, enabling direct comparison with CFT predictions.
  • The candidate boundary description suggests that stabilizer moments in critical chains may be systematically computed using orbifold CFT techniques, though the endpoint matching remains unresolved.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The basis dependence of the decay exponent implies that in holographic settings, the bulk dual of stabilizer moments must depend on the choice of operator basis, not just the entanglement wedge; this may constrain the dictionary between magic and geometry in AdS/CFT.
  • The ability of an on-site rotation to change the leading exponent suggests that other local basis changes (e.g., depth-one Clifford circuits) could similarly alter the long-distance behavior of stabilizer moments, which could be tested in quantum simulators.
  • The open kink hypothesis for higher Rényi indices (α>4) indicates a possible crossover from identity-dominated to power-law decay that may require larger-scale numerical simulations or analytic bounds to resolve.
Original abstract

Stabilizer R\'enyi moments depend on a choice of operator basis and need not be determined by entanglement spectra. In this paper, we study their spatial decay in critical quantum chains described by two-dimensional conformal field theories. We derive an exact entanglement baseline for reductions of pure stabilizer states and a symmetry selection rule for the charged contribution at fixed support. An exact Ising example shows that an on-site rotation changes the full decay exponent while preserving every subsystem entanglement spectrum. For growing Ising intervals at index two, we obtain an exact Pfaffian representation and compare the charged amplitude with a candidate conformal boundary calculation. The data support this prediction, but its identification with the lattice limit still requires endpoint matching and control of the order of limits.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-21: 1 request, 4,851 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.95 · human participants 0.01 · controlled comparison 0.54 · randomized assignment 0.01 · states quantitative result 0.24 · compares with prior results 0.92 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

Stabilizer R\'enyi moments depend on a choice of operator basis and need not be determined by entanglement spectra. In this paper, we study their spatial decay in critical quantum chains described by two-dimensional conformal field theories. We derive an exact entanglement baseline for reductions of pure stabilizer states and a symmetry selection rule for the charged contribution at fixed support. An exact Ising example shows that an on-site rotation changes the full decay exponent while preserving every subsystem entanglement spectrum. For growing Ising intervals at index two, we obtain an exact Pfaffian representation and compare the charged amplitude with a candidate conformal boundary calculation. The data support this prediction, but its identification with the lattice limit still requires endpoint matching and control of the order of limits.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Stabilizer R\'enyi moments depend on a choice of operator basis and need not be determined by entanglement spectra. background 0.88 0.16 0.43 asserted 0.93
s1 In this paper, we study their spatial decay in critical quantum chains described by two-dimensional conformal field theories. contribution 0.98 0.84 0.05 asserted 0.80
s2 We derive an exact entanglement baseline for reductions of pure stabilizer states and a symmetry selection rule for the charged contribution at fixed support. result 0.61 0.96 0.12 asserted 1.00
s3 An exact Ising example shows that an on-site rotation changes the full decay exponent while preserving every subsystem entanglement spectrum. result 0.99 0.87 0.37 asserted 1.00
s4 For growing Ising intervals at index two, we obtain an exact Pfaffian representation and compare the charged amplitude with a candidate conformal boundary calculation. result 0.62 0.94 0.08 asserted 1.00
s5 The data support this prediction, but its identification with the lattice limit still requires endpoint matching and control of the order of limits. limitation 0.95 0.78 0.11 hedged possibility 0.67

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.99.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. Growing equal intervals in infinite critical Ising at index two. (a) The charged coeffi￾cient (5.7), with the conditional CFT amplitude shown as a dashed line. (b) The rescaled linear neutral coefficient (A.14), whose near constancy supports A1 ∝ 1/ℓ. Solid segments join adjacent sizes. overlap at every finite cutoff, since such overlaps can still vanish in the limit. The third is needed because convergence only at fixed r/ℓ would not justify the far field first order of limits in (5.2). The value 1/4 here is an amplitude multiplying η 1/2 , not the rotated Ising exponent in (3.8)
Figure 2. Separated regions in infinite critical Ising. (a) The charged ratio for two length-two intervals follows η α/4 . (b) For one site intervals, the charged and single sheet neutral powers are α/4 and 1, the grey curve is their fixed support minimum, and above α = 4 the full ratio fits mix these lattice terms. In both panels, the α = 1 entries show the full R´enyi-2 mutual information, rather than a charged component or the singular functional Mcα at index one. probe different charged dimensions. The latter are taken at the Fateev–Zamolodchikov point [30], H = − X j X d−1 n=1 (Z †…
Figure 3. Fixed support exponents for the clock models and the Blume–Capel control. The dotted guides use the exact charged CFT dimensions in (3.3), and the grey control points are mean full ratio powers over separation windows. The Ising charged points are fits to the six smallest cross-ratios, η ≤ 1/576, while the interacting clock points use their full separation windows
Figure 4. Local logarithmic slopes of JM(ℓ) in infinite critical Ising for M = S2 (triangles) and M = Mc2 (circles), plotted at 1/(ℓ + 1/2) for ℓ = 2, 3, 4, 5. The dotted lines are separate linear fits, and the filled symbols at the origin are their extrapolations rather than computed sizes, giving 0.1267 for the R´enyi-2 entanglement and 0.1264 for the stabilizer series, to be compared with c/4 = 1/8. The two finite size drifts have opposite signs. For the non-Weyl control, we use HBC = − P j S z j S z j+1−λ P j S x j +D P j (S z j ) 2 at D = 0, λ = 1.326, and L = 12, near the Ising critical…

Figures reproduced from arXiv:2610.27596 by the authors.

References

Discussion

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