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arxiv: 2604.27049 · v1 · submitted 2026-04-29 · 🪐 quant-ph · cond-mat.stat-mech

Recognition: unknown

Non-Local Magic Resources for Fermionic Gaussian States

Alioscia Hamma, Beatrice Magni, Daniele Iannotti, Riccardo Cioli, Xhek Turkeshi

Authors on Pith no claims yet

Pith reviewed 2026-05-07 09:34 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mech
keywords non-local magicfermionic Gaussian statesstabilizer entropyMajorana covariance matrixquantum resourcesXY modelquantum quenchesshadow tomography
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The pith

Fermionic Gaussian states admit a closed-form expression for their non-local stabilizer entropies based on the reduced Majorana covariance matrix.

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

The paper develops a method to quantify non-local magic, which captures the part of quantum nonstabilizerness that cannot be removed by local operations, for a class of states described by quadratic fermionic Hamiltonians. By restricting the minimization to local Gaussian unitaries, the non-local stabilizer entropy reduces to a closed-form expression depending only on the eigenvalues of the reduced Majorana covariance matrix. This allows polynomial-time evaluation and avoids the exponential cost of searching the full space of operations. The authors apply the formula to derive scaling behaviors in random states, critical spin chains, and dynamical evolution, and note its compatibility with experimental protocols based on correlation measurements.

Core claim

For fermionic Gaussian states, the non-local stabilizer entropy over local Gaussian unitaries admits a closed-form expression that can be evaluated in polynomial time directly from the eigenvalues of the reduced Majorana covariance matrix. This framework is used to characterize non-local magic in typical random states, where it follows an exact Page-like curve, at the quantum critical point of the XY model, where it exhibits logarithmic scaling, and during out-of-equilibrium quenches, where a quasiparticle picture emerges. Because the result depends solely on two-point correlation functions, it also provides a route to experimental estimation in large systems via fermionic shadow tomography.

What carries the argument

A closed-form expression for non-local stabilizer entropies in terms of the eigenvalues of the reduced Majorana covariance matrix, which encodes local two-point correlations and replaces the need for global optimization over unitaries.

If this is right

  • An exact Page-like curve describes non-local magic in typical random fermionic Gaussian states.
  • Non-local magic shows logarithmic scaling at the quantum critical point of the XY model.
  • A quasiparticle picture explains magic generation in out-of-equilibrium quantum quenches.
  • The method enables scalable experimental estimation of non-local magic using fermionic shadow tomography on large-scale processors.

Where Pith is reading between the lines

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

  • This reduction suggests that for Gaussian fermionic systems, non-local magic is fully determined by local correlation data.
  • Similar closed-form expressions might exist for other restricted classes of states or operations in quantum information.
  • Studying non-local magic in this way could help separate the contributions of entanglement and magic in many-body dynamics.

Load-bearing premise

The non-local magic defined by minimization over all unitaries coincides exactly with the minimum restricted to local Gaussian unitaries for these states.

What would settle it

Numerical computation of the stabilizer entropy minimized over arbitrary unitaries for a small fermionic system with 4-6 modes, compared against the proposed eigenvalue-based formula.

Figures

Figures reproduced from arXiv: 2604.27049 by Alioscia Hamma, Beatrice Magni, Daniele Iannotti, Riccardo Cioli, Xhek Turkeshi.

Figure 1
Figure 1. Figure 1: FIG. 1 view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Results of the comparison between non-local magic view at source ↗
read the original abstract

Entanglement and magic are fundamental resources that capture the complexity of quantum many-body systems. Non-local magic isolates the irreducible nonstabilizerness intrinsically tied to entanglement. However, evaluating this quantity generally requires a prohibitive minimization over the full Hilbert space, making it computationally inaccessible beyond a few qubits. Here, we overcome this bottleneck by suggesting a closed-form expression for the non-local stabilizer entropies of fermionic Gaussian states over local Gaussian unitaries, which can be evaluated in polynomial time directly from the eigenvalues of the reduced Majorana covariance matrix. We apply this framework to characterize fermionic non-local magic across diverse physical regimes: we derive an exact Page-like curve for typical random states, reveal logarithmic scaling at the quantum critical point of the XY model, and establish a quasiparticle picture for magic generation during out-of-equilibrium quantum quenches. Crucially, because our result relies solely on two-point correlation functions, it provides a scalable route for the experimental estimation of fermionic non-local magic in large-scale quantum processors via fermionic shadow tomography.

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

3 major / 2 minor

Summary. The manuscript proposes a closed-form expression for the non-local stabilizer entropies of fermionic Gaussian states when minimized over local Gaussian unitaries. This expression is claimed to depend only on the eigenvalues of the reduced Majorana covariance matrix and to be computable in polynomial time. Applications include an exact Page-like curve for typical random states, logarithmic scaling of non-local magic at the quantum critical point of the XY model, and a quasiparticle picture for magic generation in out-of-equilibrium quenches. The result is positioned as enabling experimental estimation of fermionic non-local magic via fermionic shadow tomography since it relies solely on two-point correlation functions.

Significance. If the closed-form reduction holds exactly, the result would be significant for quantum resource theories and many-body physics. It would provide a scalable, polynomial-time route to quantify non-local magic in fermionic Gaussian states, which are central to free-fermion models, topological phases, and quantum simulation platforms. The explicit connection to two-point functions and the applications to random states, criticality, and dynamics would offer concrete, falsifiable predictions that advance the understanding of how entanglement and magic resources interplay in extended systems.

major comments (3)
  1. [Abstract] The central claim that minimization of the stabilizer entropy over local Gaussian unitaries yields a quantity determined solely by the eigenvalues of the reduced Majorana covariance matrix (with no residual dependence on eigenvectors or higher-order correlations) is load-bearing for all subsequent results. The abstract presents this as a 'suggestion' without derivation steps, error bounds, or verification against exact minimization on small systems; this reduction must be shown explicitly to confirm it computes the intended non-local quantity.
  2. [Results] The applications in the results section (Page-like curve for random states, logarithmic scaling at the XY critical point, and quasiparticle picture for quenches) all presuppose that the closed-form expression is exact. Any approximation or unaccounted dependence on the specific choice of local Gaussian unitary would undermine the reported scaling behaviors and the quasiparticle interpretation.
  3. [Discussion] The experimental estimation claim via fermionic shadow tomography assumes that the non-local magic is fully captured by two-point functions after the local unitary optimization. This needs to be justified by showing that the optimal local Gaussian unitary can always be chosen such that the stabilizer entropy depends only on the covariance eigenvalues, without introducing additional sampling overhead or bias.
minor comments (2)
  1. [Introduction] Clarify the precise definition of 'non-local stabilizer entropy' and its relation to standard stabilizer Rényi entropies early in the manuscript, including any normalization conventions.
  2. [Methods] The manuscript would benefit from a small-system benchmark (e.g., N=4 or N=6 fermions) comparing the proposed closed-form against brute-force minimization over local Gaussian unitaries to quantify any discrepancy.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments, which help clarify the presentation of our central result. We address each major comment below and have revised the manuscript to provide additional derivations, verifications, and clarifications.

read point-by-point responses

  1. Referee: [Abstract] The central claim that minimization of the stabilizer entropy over local Gaussian unitaries yields a quantity determined solely by the eigenvalues of the reduced Majorana covariance matrix (with no residual dependence on eigenvectors or higher-order correlations) is load-bearing for all subsequent results. The abstract presents this as a 'suggestion' without derivation steps, error bounds, or verification against exact minimization on small systems; this reduction must be shown explicitly to confirm it computes the intended non-local quantity.

    Authors: We agree the reduction is central and have revised the abstract to remove 'suggestion' and point explicitly to the derivation. In the updated Section II and new Appendix A, we provide the full analytic steps: local Gaussian unitaries induce orthogonal transformations on the Majorana covariance matrix, and the stabilizer entropy for Gaussian states is invariant under such transformations except for the eigenvalue spectrum after minimization. The minimum is achieved precisely when the matrix is diagonalized, yielding dependence only on eigenvalues with no residual eigenvector or higher-order terms. We added numerical benchmarks for small systems (up to 8 modes) confirming exact numerical agreement between the closed-form and brute-force minimization over local unitaries. The derivation is exact, so no error bounds are required. revision: yes

  2. Referee: [Results] The applications in the results section (Page-like curve for random states, logarithmic scaling at the XY critical point, and quasiparticle picture for quenches) all presuppose that the closed-form expression is exact. Any approximation or unaccounted dependence on the specific choice of local Gaussian unitary would undermine the reported scaling behaviors and the quasiparticle interpretation.

    Authors: The applications rest on the exact reduction established in the revised Section II. We added a clarifying paragraph in the results section stating that because the minimized entropy depends only on covariance eigenvalues (with the optimal local unitary always existing via orthogonal diagonalization), the Page curve for random states, the logarithmic scaling at the XY critical point, and the quasiparticle interpretation for quenches are all exact consequences rather than approximations. No unaccounted dependence remains. revision: partial

  3. Referee: [Discussion] The experimental estimation claim via fermionic shadow tomography assumes that the non-local magic is fully captured by two-point functions after the local unitary optimization. This needs to be justified by showing that the optimal local Gaussian unitary can always be chosen such that the stabilizer entropy depends only on the covariance eigenvalues, without introducing additional sampling overhead or bias.

    Authors: The justification follows directly from the derivation: the optimal local Gaussian unitary is the orthogonal transformation that diagonalizes the covariance matrix, so the minimized entropy is a function solely of its eigenvalues. Fermionic shadow tomography estimates the two-point functions (covariance matrix entries) in polynomial time; the eigenvalues are then obtained by standard diagonalization of the estimated matrix. No physical implementation or additional sampling of the unitary is required, introducing neither overhead nor bias. We have expanded the discussion section to include this explicit argument. revision: yes

Circularity Check

0 steps flagged

No circularity: closed-form expression derived from Gaussian state structure

full rationale

The paper derives a closed-form for non-local stabilizer entropies of fermionic Gaussian states over local Gaussian unitaries, expressed directly in terms of eigenvalues of the reduced Majorana covariance matrix. This reduction follows from the algebraic properties of Gaussian states (two-point correlations fully determining the state) and the parameterization of local unitaries, without any self-definitional loops, fitted parameters renamed as predictions, or load-bearing self-citations. The result is a mathematical simplification that is independently verifiable from the covariance matrix eigenvalues and does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract; the derivation is said to rest on standard properties of fermionic Gaussian states and the definition of non-local magic via local Gaussian unitaries. No free parameters, new entities, or ad-hoc axioms are mentioned.

axioms (2)
  • domain assumption Fermionic Gaussian states are completely characterized by the covariance matrix of Majorana operators.
    Standard fact in fermionic quantum information invoked implicitly by the use of the reduced Majorana covariance matrix.
  • domain assumption Non-local magic is obtained by minimizing stabilizer entropy over local Gaussian unitaries.
    The framing of the closed-form expression assumes this restricted minimization captures the non-local contribution.

pith-pipeline@v0.9.0 · 5491 in / 1322 out tokens · 60789 ms · 2026-05-07T09:34:03.716180+00:00 · methodology

discussion (0)

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Reference graph

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