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REVIEW 4 major objections 4 minor 1 cited by

Explicit Pfaffian Formula for Amplitudes of Fermionic Gaussian Pure States in Arbitrary Pauli Bases

T0 review · 4 major / 4 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read A single Pfaffian formula computes Gaussian-state amplitudes in any Pauli basis, with a recursion that handles large qubit systems.

desk verdict A genuinely new and likely correct Pfaffian formula for even L, but the odd-L extension rests on an unproven identity and an under-specified phase convention; referee it, but insist on a real proof and small-L checks. read the letter →

arxiv 2502.04857 v3 pith:JSGDV7KI submitted 2025-02-07 quant-ph cond-mat.str-elmath-phmath.MP

M. A. Rajabpour , M. A. Seifi Mirjafarlou , Reyhaneh Khasseh This is my paper · ORCID
classification quant-phcond-mat.str-elmath-phmath.MP MSC 15A1581P68
keywords fermionicGaussianstatesPfaffianPaulibasisamplitudesquantumtomographyShannon-Renyientropyformationprobabilitypost-measuremententanglementmatchgatecircuits
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper establishes an explicit Pfaffian formula for the amplitude of any product-state configuration in a local Pauli basis, for a fermionic Gaussian pure state on $L$ qubits. The formula expresses an amplitude as a trigonometric prefactor times a single Pfaffian of an $L\times L$ matrix, or an $(L+1)\times(L+1)$ matrix when $L$ is odd, so an individual amplitude costs polynomial time and systems with $L>1000$ become numerically accessible. A second result gives a recursive relation that builds $L$-qubit amplitudes from $2$-qubit and $(L-2)$-qubit amplitudes, enabling incremental computation. The authors apply the formulas to formation probabilities, Shannon-R\'enyi entropies, tomography log-likelihoods, and post-measurement entanglement entropy, and they extract conformal-field-theory scaling exponents in the transverse-field Ising chain.

What carries the argument

The object carrying the argument is the spin-dependent antisymmetric matrix $R_S(\phi,\theta,\alpha)$, whose entries are given in Eq. (8): $$$R_S^{{nm}}$=r_{nm}$e^{{i(\phi_n+\phi_m)}}$+(-1)^{n+m}(-1)^{(\bar{s}_n+\bar{s}_m)/2}\$tan^{{\bar{s}}$_n}\frac{\theta_n}{2}\$tan^{{\bar{s}}$_m}\frac{\theta_m}{2}$$ for even $L$, with the odd-$L$ case obtained by adding a zero row and column under the ancilla conditions. Taking the Pfaffian of this matrix converts the exponentially many terms in the basis expansion into one algebraic expression. The proof machinery is the classical theorem that expresses a Pfaffian with shifted weights as a sum over submatrix Pfaffians, extended by Theorem 3 to odd-dimensional antisymmetric matrices with an appended zero row and column.

What would settle it

For a 3-qubit Gaussian pure state with a random antisymmetric $R$, expand the state vector exactly in a generic Pauli basis and compare one amplitude with Eq. (7) using the stated ancilla rule; a mismatch at generic $\theta$ values would falsify the odd-$L$ claim. The same check can be done purely symbolically: expand the right-hand side of Theorem 3 for a random $3\times3$ antisymmetric matrix and compare it with the direct Pfaffian definition.

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Extended reading notes

Core claim

The central discovery is Theorem 1: for a Gaussian pure state $|R,0\rangle$ written through an antisymmetric matrix $R$, the amplitude in the local Pauli basis $(\phi_j,\theta_j,\alpha_j)$ is $$a_S(R,\phi,\$\theta$,\$\alpha$)=\frac{(-1)^{L(1-\bar{s}_1)/2}\,(\sqrt{2})^{L\bmod 2}}{N_R}\,$e^{{-i\sum_{j\in S_-}}$\alpha_j}\prod_{j\in S_+}\cos\frac{\theta_j}{2}\prod_{j\in S_-}\sin\frac{\theta_j}{2}\,\operatorname{pf}R_S,$$ where the matrix $R_S$ is built from $R$ by phase shifts in $\phi$ and by diagonal trigonometric terms determined by the spin configuration $S$. For even $L$ this is the complete statement; for odd $L$ the paper appends a zero row and column with $\bar{s}_{L+1}=\bar{s}_1$, $\theta_{L+1}=\pi/2$, and $\alpha_{L+1}=0$. The proof reduces the amplitude sum to a Pfaffian expansion, using an odd-dimensional generalization of the classical Pfaffian theorem (Theorem 3) in the odd case. The companion Theorem 2 is a Pfaffian-based recursion relating the $L$-qubit amplitude to products of amplitudes for two qubits and $L-2$ qubits, with shifted angles. If these statements hold, every individual amplitude, and hence every single-outcome probability, is available in polynomial time for Gaussian pure states.

Load-bearing premise

The odd-qubit case of Theorem 1 depends entirely on a new odd-dimensional extension of the classical Pfaffian expansion together with a specific ancilla choice (same spin direction as qubit 1, $\theta_{L+1}=\pi/2$, $\alpha_{L+1}=0$); if that extension or that choice is wrong, all odd-$L$ amplitudes from the formula are wrong.

Editorial extensions

If this is right

  • Formation probabilities in any Pauli basis become directly computable from the determinant formula (11), without summing over the full Hilbert space.
  • Shannon-R\'enyi entropies and global entanglement, which require many probabilities, can be computed or optimized in regimes previously out of reach; the authors emphasize system sizes $L>1000$ for individual amplitudes.
  • Negative log-likelihood functions for quantum tomography of Gaussian pure states can be evaluated with explicit amplitude expressions, enabling parameter estimation from projective measurements in experimentally natural bases.
  • Post-measurement entanglement entropy can be extracted for arbitrary measurement bases; in the critical transverse-field Ising chain the paper finds power-law decay and scaling dimensions $\Delta_1=1/2$ in the $\sigma_z$ basis and $\Delta_1=2$ in the $\sigma_x$ basis, consistent with CFT.
  • The recursion of Theorem 2 lets amplitudes for larger systems be built from smaller ones with adjusted angles, supporting incremental simulation of matchgate circuits and parallelized updates.

Reading between the lines

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

  • One implication the authors only touch on: for odd $L$, a robust odd-dimensional Pfaffian theorem would make the same mechanism available for Gaussian mixed states with unpaired Majorana modes, not just pure states.
  • A direct numerical check of Theorem 3 on random antisymmetric matrices of odd size, independent of any quantum state, would be the cheapest way to certify the ancilla construction.
  • The recursive relation could be restated as an incremental update rule for matchgate circuits that updates amplitudes two qubits at a time, potentially avoiding full Pfaffian recomputation; the paper suggests this but does not develop the circuit-level algorithm.
  • For global entanglement, the explicit angle dependence turns the required maximization over bases into a continuous optimization over the $\theta_j,\phi_j$ parameters of one matrix $R_S$, which is likely far more tractable than the exponential enumeration implied by the definition.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper claims an explicit Pfaffian formula (Theorem 1) for the amplitudes of fermionic Gaussian pure states in arbitrary local Pauli bases, expressed in Eqs. (7)-(8), with an auxiliary row/column and an ancilla prescription for odd qubit number L. It also states a recursive amplitude relation (Theorem 2), a determinant formula for formation probabilities (Eq. (11)), and applies the framework to compute post-measurement entanglement entropy in the critical transverse-field Ising chain, extracting decay exponents that are compared with CFT expectations. If valid, the formula evaluates individual amplitudes in polynomial time for large L, which would matter for tomography, formation probabilities, Shannon-Renyi entropies, and post-measurement entanglement calculations.

Significance. The central even-L formula is plausible and valuable: it reduces to known Pfaffian amplitudes in the computational basis, gives an explicit angle-dependent extension to arbitrary Pauli bases, and yields a simple probability formula that can be evaluated for systems with L > 1000. The recursive Theorem 2, if correct, is a useful additional structural result. The paper does not supply machine-checked proofs or numerical verification code, and the odd-L case rests on a new Pfaffian identity whose proof is only sketched. The application to post-measurement entanglement entropy is interesting but the comparison with CFT is a fit of Delta_1 rather than a parameter-free prediction. Overall, the contribution is significant if the missing proofs and conventions are supplied.

major comments (4)
  1. [Appendix B, Theorem 3] All odd-L amplitudes in Eq. (7) depend on the ancilla prescription (sbar_{L+1}=sbar_1, theta_{L+1}=pi/2, alpha_{L+1}=0) and on Theorem 3, but Theorem 3 is not proved: the text only states that it can be proven by Lieb's original approach or by a Berezin integral representation, and the preceding proof of Theorem 1 uses 'Direct inspection shows' and 'one can reach' at the key combinatorial step. Because the Pfaffian of an odd-dimensional matrix R vanishes identically, this is not a decorative extension; a sign error in lambda_{L+1}=1 or in the ancilla weights would change every odd-L amplitude. Please provide a complete proof of Theorem 3 and of the ancilla reduction, or an independent numerical check of Eq. (7) for L=1,3,5 with random R and a range of angles.
  2. [Section III, Eq. (7) vs Eq. (2)] The phase and normalization convention for a_S is under-specified. For R=0 and L=1, Eq. (7) gives an up-spin amplitude proportional to sin(theta/2) (up to the sqrt(2) factor from L mod 2 and the overall sign), whereas applying the unitary U of Eq. (1) to the vacuum |0> in the mapping of Eq. (2) gives cos(theta/2) as the coefficient of the corresponding up basis vector. The paper does not state whether a_S is <S|U|psi>_z, <S|psi>_{(phi,theta,alpha)}, or some other convention, nor how the factors (-1)^{L(1-sbar_1)/2} and sqrt(2)^{L mod 2} are fixed. Please specify the convention and verify the R=0, L=1,2 limits explicitly.
  3. [Section III, Eq. (8)] As written, R_S,nm is not antisymmetric for all n,m: the first term r_nm is antisymmetric, but the second term is symmetric in n and m, so without an explicit n<m (or equivalent antisymmetrization) the Pfaffian in Eq. (7) is not defined. Appendix C states 'for m>n' only for the equivalent matrix M, not for Eq. (8). Please state the ordering convention in the main theorem.
  4. [Appendix E, Theorem 2] The proof of Theorem 2 is only a verbal description of a Pfaffian decomposition ('can be derived directly from Theorem 1 by employing the fundamental properties of the Pfaffian'), and the explicit formulas in Table III are presented without derivation. Since Theorem 2 is advertised as a second main result and as the basis for scalable incremental computations, the derivation should be completed or made machine-checkable.
minor comments (4)
  1. [General] The sentence 'in which nm is the complement of the nm' after Theorem 2 is garbled and should be rewritten.
  2. [References] Reference [24] lists 'Phys. Rev. A 64, 064412 (2024)'; volume 64 cannot be a 2024 volume. Please correct the bibliographic data.
  3. [Appendix A] The construction of R' in Eq. (A2) is hard to follow; a small worked example of the base-configuration change would clarify the sign conventions.
  4. [Section V] The comparison with CFT is not a parameter-free test: Delta_1 in Eq. (18) is extracted from the same numerical data, so the agreement only confirms the power-law form and the fitted exponent values. Please state this explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 1 is an algebraic identity derived from an independent Gaussian-state representation; the CFT comparison is a consistency check.

full rationale

The central derivation is self-contained given the standard fermionic-Gaussian representation |R,0> = (1/N_R) exp(1/2 Σ a_i r_ij a_j)|0> borrowed from ref [24]. That representation is an independent published result and a standard Bogoliubov form; it is not a claim whose content is the target amplitude formula, so the self-citation is not load-bearing circularity. Theorem 1 is obtained by expanding U(φ,θ,α)|ψ>_z, grouping terms by Pfaffian submatrices, and applying Lieb's Pfaffian theorem (refs [48,49]) to reassemble the trigonometric sums; the matrix R_S in Eq. (8) is defined from r_nm and the Pauli angles rather than from the amplitudes themselves, so equality (7) is a genuine identity rather than a restatement of its own definition. Theorem 2 follows algebraically from Pfaffian row-expansion properties, and Theorem 3 (odd L) is a separate Pfaffian identity justified by a Berezin-integral argument; any gap in that proof would be a correctness issue, not a circular one. The Section V comparison with CFT uses Eq. (17)-(18) with Δ1 extracted numerically from the same entanglement data; that is a consistency check and does not feed back into the proof of Theorem 1. No fitted parameter is renamed as a prediction, and no self-citation is invoked to forbid alternatives.

Assumptions & free parameters 1 free parameters · 3 assumptions · 1 invented entities

The core formula has no fitted constants. It rests on the Gaussian-state representation from ref [24] (self-cited), Lieb's theorem, and an ad hoc ancilla for odd L. The only fitted number in the paper is the scaling dimension Delta_1 in the application section.

free parameters (1)
  • Delta_1 (scaling dimension) = 1/2 (sigma_z), 2 (sigma_x), 1, 1/2 for Table I configurations
    Extracted by fitting the numerical decay of post-measurement entanglement entropy to d^{-eta(alpha)} in Section V; it is not predicted independently by the paper and is used only in the application, not in Theorem 1.
assumptions (3)
  • domain assumption Any fermionic Gaussian pure state can be represented as |R,C> = (1/N_R) exp(1/2 sum a_i r_ij a_j)|C> with normalization det(I+R^dagger R)^{1/4}, and the base configuration can be changed via the procedure in Appendix A (from ref [24]).
    Invoked in Section II and Appendix A; the representation and base-change formulas are taken from prior work by two of the current authors ([24]) and are not re-derived here.
  • standard math Lieb's theorem on Pfaffians (and its odd-dimensional generalization, Theorem 3) correctly sums the trigonometric coefficients into a single Pfaffian.
    Used in Appendix B to obtain Eq (7). The generalization to odd L is new to this paper and proven only in outline via Berezin integrals.
  • ad hoc to paper The odd-L amplitude requires an ancillary qubit with sbar_{L+1}=sbar_1, theta_{L+1}=pi/2, alpha_{L+1}=0.
    Introduced after Eq (8) and in Appendix B; no external validation is provided beyond the claimed proof.
invented entities (1)
  • Ancillary qubit for odd L
    purpose: Adds a zero row/column so the Pfaffian matrix has even dimension for odd-qubit systems.
    A calculational device, not a physical particle; its correctness rests entirely on Theorem 3's proof outline.

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Cite this review

Pith. "Pith review of Explicit Pfaffian Formula for Amplitudes of Fermionic Gaussian Pure States in Arbitrary Pauli Bases." pith.science (2026). https://pith.science/paper/JSGDV7KI

@misc{pith2026250204857,
  author       = {Pith},
  title        = {Pith review of: Explicit Pfaffian Formula for Amplitudes of Fermionic Gaussian Pure States in Arbitrary Pauli Bases},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JSGDV7KI}},
  note         = {Machine review of arXiv:2502.04857}
}
read the original abstract

The explicit computation of amplitudes for fermionic Gaussian pure states in arbitrary Pauli bases is a long-standing challenge in quantum many-body physics, with significant implications for quantum tomography, experimental studies, and quantum dynamics. These calculations are essential for analyzing complex properties beyond traditional measures, such as formation probabilities, global entanglement, and entropy in non-standard bases, where exact and computationally efficient methods remain underdeveloped. In addition to these physical applications, having explicit formulas is crucial for optimizing negative log-likelihood functions in quantum tomography, a key task in the NISQ era. In this work, we present an explicit Pfaffian formula (Theorem 1) for determining these amplitudes in arbitrary Pauli bases, utilizing a matrix whose structure reflects the qubit parity. Additionally, we introduce a recursive relation (Theorem 2) that connects amplitudes for systems with varying qubit numbers, enabling scalable computations for large systems. Together, these results provide a versatile framework for studying global entanglement, Shannon-R\'enyi entropies, formation probabilities, and performing efficient quantum tomography, thereby significantly expanding the computational toolkit for analyzing complex quantum systems. Finally, we utilize our formalism to determine the post-measurement entanglement entropy, reflecting how local measurements alter entanglement, and compare the outcomes with conformal field theory predictions.

Figures

Figures reproduced from arXiv: 2502.04857 by the authors.

Figure 1
Figure 1. FIG. 1: Schematic of a one-dimensional chain with [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. displays the behavior of E (α,SB) A1 (d) for the measurement outcomes |B1⟩ = |+⟩ ⊗LB1 and |B2⟩ = |+⟩ ⊗LB2 , in both the σ z and σ x bases. In both cases, the entanglement entropy exhibits a clear power-law de￾cay with distance, in accordance with Eq. (17). The ex￾tracted scaling dimensions are ∆1 = 1/2 for the σ z basis and ∆1 = 2 for the σ x basis. A broader set of measurement outcomes, along with their correspondi… view at source ↗

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Matrix Elements of Fermionic Gaussian Operators in Arbitrary Pauli Bases: A Pfaffian Formula

    quant-ph 2025-06 conditional novelty 5.0 of 10

    Every matrix element of a fermionic Gaussian operator between arbitrary Pauli product states is expressed as a single Pfaffian of a 2L by 2L kernel with explicitly tabulated sign matrices.

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Reviewed August 8, 2026 · model on record in the stance chip above.