Relative entropy of single-mode squeezed states in Quantum Field Theory

Arthur F. Vieira; Itzhak Roditi; Marcelo S. Guimaraes; Silvio P. Sorella

REVIEW 3 major objections 5 minor 52 references

Relative entropy of single-mode squeezed states in Quantum Field Theory

T0 review · 3 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read A closed-form Araki-Uhlmann relative entropy for squeezed states in a free massive scalar QFT is derived and shown to be proportional to the smeared Pauli-Jordan distribution.

desk verdict Clean extension of the coherent-state relative-entropy computation, but the central modular identity rests on an unproved and likely false assumption that S_f lies in the wedge algebra. read the letter →

arxiv 2504.13148 v2 pith:AWQQWMIY submitted 2025-04-17 hep-th math-phmath.MP

Marcelo S. Guimaraes , Itzhak Roditi , Silvio P. Sorella , Arthur F. Vieira This is my paper · ORCID

classification hep-thmath-phmath.MP MSC 81T0546L6081P45 PACS 03.70.+k

keywords Araki-Uhlmannrelativeentropysingle-modesqueezedstateTomita-TakesakimodulartheoryBisognano-WichmanntheoremPauli-Jordandistributionwedgeregionfreescalarquantumfieldflow

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

The paper derives an exact, closed-form expression for the Araki-Uhlmann relative entropy between a single-mode squeezed vacuum state and the ordinary vacuum in a free massive real scalar quantum field theory, restricted to a wedge region of 1+1 Minkowski spacetime. The result is that this entropy is proportional to the smeared Pauli-Jordan distribution, the same causal, antisymmetric two-point object that controls the coherent-state relative entropy. If the derivation is correct, the entropy inherits all the physically expected properties without approximation: it is positive, it grows with the size of the region, and it falls as the field mass increases. That matters because relative entropy is the rigorous information-theoretic measure of distinguishability in algebraic QFT, and this gives an analytic, non-numerical handle on it for a genuinely nonlinear family of states.

What carries the argument

The load-bearing mechanism is the Tomita-Takesaki modular theory for the wedge algebra. The relative modular operator $\Delta_{\Psi|\Omega}$ is identified with the vacuum modular operator $\Delta_\Omega$ through $\Delta_{\Psi|\Omega}^{is}=\Delta_\Omega^{is}$, a consequence of unitarity of $S_f$ and the algebra automorphism it generates. The Bisognano-Wichmann theorem then identifies the vacuum modular flow with a Lorentz boost, so $\Delta_\Omega^{is} S_f \Delta_\Omega^{-is}=S_{f_s}$ with $f_s(x)=f(\Lambda_{-s}x)$. The remaining computation is the two-point function $\langle\Omega|S_f^\dagger S_{f_s}|\Omega\rangle$, evaluated in closed form as $C_f C_{f_s}(1+4\alpha_{f_s}\gamma_f\langle f|f_s\rangle^2)^{-1/2}$; differentiating at $s=0$ produces the Pauli-Jordan term.

What would settle it

Compare the two sides of Eq. (40) directly: evaluate $\langle\Omega|S_f^\dagger\Delta_\Omega^{is}S_f|\Omega\rangle$ using the vacuum modular Hamiltonian on the wedge and evaluate $\langle\Omega|S_f^\dagger S_{f_s}|\Omega\rangle$ using the Bisognano-Wichmann boost, for a profile $f$ supported strictly inside the wedge. Any nonzero difference at small $s$ would show $\Delta_{\Psi|\Omega}^{is}\neq\Delta_\Omega^{is}$ and invalidate the central formula; a lattice discretization of the free massive scalar provides a practical setting for this comparison.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is the identity $$S(\Psi|\$\Omega$)=-\frac{\$alpha_f^{2}$ $C_f^{2}$}{(1-4\$alpha_f^{2}$\|f\|^4)^{3/2}\|f\|^2}\,\Delta_{PJ}\!\left(f,f'_s\big|_{s=0}\right),$$ where $\Psi=S_f|\Omega\rangle$ is the single-mode squeezed state with $S_f=e^{i(a_f^2+(a_f^\dagger)^2)}$, $\|f\|^2$ is the Lorentz-invariant norm of the smearing function, and $\Delta_{PJ}$ is the smeared Pauli-Jordan commutator function. The argument shows that the relative modular operator of $(\Psi,\Omega)$ coincides with the vacuum modular operator $\Delta_\Omega$, so the Bisognano-Wichmann boost can be applied and the entropy reduces to a derivative of the two-point function $\langle\Omega|S_f^\dagger S_{f_s}|\Omega\rangle$. The closed form then follows from the squeezed-state algebra and the disentangling formula for $S_f$.

Load-bearing premise

The whole derivation hinges on the squeezing operation being generated inside the wedge algebra, so that the relative modular flow coincides with the vacuum's, whereas the paper proves only that it maps the algebra to itself and never shows $S_f$ lies in the algebra.

Editorial extensions

If this is right

For any test function $f$ supported in the right wedge, the formula gives the exact Araki-Uhlmann relative entropy of the squeezed state, with no approximation or cutoff.
The entropy is strictly positive, inheriting the sign of the smeared Pauli-Jordan term.
The entropy increases when the wedge region is enlarged, since the smeared commutator term grows with the support of $f$.
The entropy decreases as the mass $m$ grows, following the mass dependence of the Bessel functions in $\Delta_{PJ}$.
Because $S(\Psi|\Omega)$ is a fixed multiple of the coherent-state relative entropy, it inherits the representation of that entropy as an integral of the smeared energy-momentum tensor.

Reading between the lines

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

Rearranging Eq. (41) shows the prefactor equals $\sinh^2(2\|f\|^2)/(4\|f\|^6)$, so, up to the Pauli-Jordan smearing, the entropy is governed by this positive function of the squeezing strength; monotonic growth with squeezing follows directly, although the paper does not state it.
The same modular-flow strategy should extend to two-mode squeezing with a kernel $f(p,q)$; the two-point function would become a determinant over the kernel's modes, and the relative entropy should again reduce to a Pauli-Jordan smearing with a mode-dependent prefactor.
Because the derivation is algebraic and exact, a lattice discretization of a free scalar field on a chain with a squeezed mode could check Eq. (41) in the continuum limit; this would also test the inner-automorphism step that the paper leaves open.
The proportionality to the coherent-state entropy suggests that, for all single-mode Gaussian states, the relative entropy to the vacuum is not an independent quantity but a one-parameter rescaling of the coherent-state result, so any measurement sensitive to the vacuum commutator would see the same functional form.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Quick take: this is a clean, readable paper that does exactly what it says: it carries the coherent-state calculation in Casini-Grillo-Pontello over to single-mode squeezed states and lands on a closed form proportional to the smeared Pauli-Jordan distribution. The algebra in the main text is careful, and the final expression is the kind of thing people will want to have on record if it is right. But the pivotal move in Eq. (28), where Δ_{Ψ|Ω} is set equal to Δ_Ω, is not justified. The argument in Eq. (30) silently requires S_f to lie in the wedge algebra M; Appendix B shows at most that conjugation by S_f maps Weyl operators to Weyl operators, and even that is doubtful because the map g → g+ηf with complex η does not preserve the Weyl phase. For a free field, S_f is a Bogoliubov transformation of the type that is generally not in the local algebra; the modular group itself is a counterexample to the idea that such exponentials are inner. So the central identity is unsupported, and formula (41) does not follow from the derivation.

The second issue is the series in Eq. (35): they sum a power series with a finite radius of convergence and present the closed form (36) without comment. That might be fixable via analytic continuation, and I want to note that the worry about the denominator in (41) going imaginary is not right—plug in α_f and you get sech^3(2||f||²), which is fine for all ||f||². So the series issue is a presentation gap, not a fatal one.

What the paper does well: the organization is solid, the notation is consistent, and the reduction to the coherent-state result in Eq. (44) is a nice sanity check. If the modular identity can be proven by other means—for instance by showing S_f is in the natural cone, or by a direct computation—the final result is plausibly correct. But as written, the load-bearing assumption is a domain issue that is never addressed.

My call: it is worth sending to a referee who knows modular theory, because the subtlety is real and the result is interesting. But it should not be accepted as is; it needs either a proof of S_f ∈ M or a different derivation of Eq. (41).

Referee Report

3 major / 5 minor

Summary. The manuscript claims a closed-form analytic expression for the Araki-Uhlmann relative entropy between a single-mode squeezed vacuum state Ψ = S_f|Ω> and the vacuum |Ω> in a free massive scalar QFT on the right Rindler wedge. The derivation invokes Tomita-Takesaki modular theory and Bisognano-Wichmann covariance, asserts the identity Δ_{Ψ|Ω}^{is} = Δ_Ω^{is}, reduces the entropy to a two-point function ⟨Ω|S_f† S_{f_s}|Ω⟩, evaluates this two-point function in closed form, and obtains Eq. (41), an expression proportional to the smeared Pauli-Jordan distribution. The paper then claims that the entropy is positive, increases with the size of the region, and decreases with mass.

Significance. If correct, the result would be a valuable explicit example of an analytically computable Araki-Uhlmann relative entropy for a non-coherent class of states, extending known coherent-state formulas with no free parameters and connecting squeezing, modular theory, and the Pauli-Jordan distribution. The paper is clearly organized and the general strategy is attractive. However, there are two load-bearing problems: the modular-theoretic identity is not justified, and the two-point function used in the derivation is demonstrably incorrect. The advertised central result is therefore not established.

major comments (3)

[III, Eqs. (28)-(31)] The proof of Δ_{Ψ|Ω}^{is} = Δ_Ω^{is} is invalid as written. In Eq. (30), applying the definition of s_{Ψ|Ω} to the vector (a S_f†)|Ψ> requires a S_f† ∈ M for all a ∈ M, i.e., S_f† ∈ M. This is never shown. Appendix B proves at most that conjugation by S_f maps Weyl generators to operators W_{g+ηf} with complex η, which need not even belong to the Weyl algebra M over real test functions; and even a genuine automorphism of M would not imply that the implementing unitary lies in M. Since Eqs. (37)-(41) replace Δ_{Ψ|Ω} by Δ_Ω at every step, the central formula is unsupported.
[IV, Eqs. (32)-(36)] The two-point function evaluation is algebraically wrong. For g = f with f real, unitarity plus the standard SU(1,1) disentangling gives ⟨Ω|S_f²|Ω⟩ = (cosh(4||f||²))^{-1/2} = 1 - 4||f||² + O(||f||⁴), whereas Eq. (36) yields approximately 1 + O(||f||⁴). For example, with ||f||² = 0.1, the exact value is about 0.962, while Eq. (36) gives about 1.000, exceeding the unitarity bound |⟨Ω|S_f²|Ω⟩| ≤ 1. This error occurs at order ||f||² in the expansion and propagates directly into the derivation of Eqs. (40)-(41).
[IV, Eqs. (35)-(36)] Independently of the algebraic error above, the power series (35) converges only for |γ_f α_g ⟨f|g⟩²| < 1/4. For g = f and for f_s with s sufficiently close to 0, this condition is satisfied only for ||f||² below roughly 1/2. The paper uses the closed form (36) and its derivatives without any restriction on ||f||² and without an analytic-continuation argument, so the transition from (40) to (41) is not justified for generic f.

minor comments (5)

[II, Eqs. (1)-(8)] The text never states whether the test functions f and g are real. The Weyl algebra (1) is defined over real smearing functions, while Eqs. (19)-(21) produce complex shifts g + ηf. The allowed test-function space should be stated precisely.
[II, Eq. (21) and Appendix B, Eq. (B16)] The coefficient called η in Eq. (21) and the coefficient called γ in Eq. (B16) are written with apparently different signs and ordering of the Hadamard and Pauli-Jordan terms, making it difficult to verify that the two expressions agree.
[II, Eq. (17)] Eq. (17) writes states as |n_f n_f⟩ but then sums over a single index n; the notation should be clarified or a normalized two-particle basis should be introduced explicitly.
[IV, Eq. (44)] The asserted monotonicity properties (increase with region size, decrease with mass) are not derived in the present paper but are imported from references [43,45]. Since Eq. (41) involves the smeared Pauli-Jordan function evaluated on f and f'_s, an explicit monotonicity argument would be needed.
[Appendix B, Eqs. (B17)-(B18)] The symbol η is used both for the complex shift in Eq. (20) and for the c-number phase in Eq. (B17); this double use is confusing and should be avoided.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation: Eq. (41) follows from an analytic computation benchmarked against external modular-theory results; same-author citations are illustrative, not load-bearing.

full rationale

The derivation of Eq. (41) is not circular: no parameter is fitted to the target quantity, and the target relative entropy is not inserted as an input. The pivotal modular-flow identity Eq. (28) is asserted as a 'remarkable relation' citing Refs. [43,44] (Casini-Grillo-Pontello and Fröb-Sangaletti), which are not by the present authors, and the subsequent use of the Bisognano-Wichmann boost action in Eq. (37) rests on an external theorem. Whether the derivation of Eq. (28) via Eq. (30) is rigorously justified depends on whether the squeezing unitary S_f lies in the wedge algebra or implements an inner automorphism; the paper only establishes an automorphism in Appendix B. That is a mathematical correctness gap, not circularity, because the conclusion is neither assumed in the premises nor obtained by renaming an input. The two-point function in Eqs. (35)-(36) is evaluated self-containedly by a series identity, and Eq. (44) merely rewrites the already-derived result in terms of the coherent-state relative entropy from the same authors' prior work [45]; that self-citation is post hoc and not load-bearing. The positivity and monotonicity statements are consequences of the explicit formula and standard properties of the Pauli-Jordan distribution. Overall, no step reduces by construction to its own inputs; the only mild concern is a non-load-bearing self-citation in Eq. (44).

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The derivation uses standard algebraic QFT theorems plus one paper-specific assumption: that S_f acts as an inner automorphism. No free parameters are fit to data and no new entities are introduced.

assumptions (5)

standard math Canonical commutation relations and Fock representation of the free massive scalar field (Appendix A, Eq. A2).
Background quantization of the field; not under dispute.
domain assumption Reeh-Schlieder theorem: vacuum is cyclic and separating for the wedge algebra M.
Needed so Tomita-Takesaki operators are well-defined (Section II, after Eq. 5).
domain assumption Bisognano-Wichmann theorem: modular flow of the vacuum on a wedge is a Lorentz boost (Eqs. 38-39).
Used to convert modular flow into a geometric boost in Eq. (40).
standard math Tomita-Takesaki modular theory and the spectral formula for relative entropy (Section III).
Framework for Araki-Uhlmann entropy.
ad hoc to paper The squeezing operator S_f implements an automorphism of M (Appendix B), and this automorphism is inner so Eq. (28) holds.
Load-bearing; the paper proves only the automorphism part, not innerness.

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Cite this review

Pith. "Pith review of Relative entropy of single-mode squeezed states in Quantum Field Theory." pith.science (2026). https://pith.science/paper/AWQQWMIY

@misc{pith2026250413148,
  author       = {Pith},
  title        = {Pith review of: Relative entropy of single-mode squeezed states in Quantum Field Theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AWQQWMIY}},
  note         = {Machine review of arXiv:2504.13148}
}

read the original abstract

Utilizing the Tomita-Takesaki modular theory, we derive a closed-form analytic expression for the Araki-Uhlmann relative entropy between a single-mode squeezed state and the vacuum state in a free relativistic massive scalar Quantum Field Theory within wedge regions of Minkowski spacetime. Similarly to the case of coherent states, this relative entropy is proportional to the smeared Pauli-Jordan distribution. Consequently, the Araki-Uhlmann entropy between a single-mode squeezed state and the vacuum satisfies all expected properties: it remains positive, increases with the size of the Minkowski region under consideration, and decreases as the mass parameter grows.

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [1]

Field expansion and commutation relations 8

work page

[2] [2]

Inner product and two-point functions 9

work page

[3] [3]

Physical interpretation and causality 10 B. Some details about the single-mode squeezed states in QFT 10 References 12 ∗ msguimaraes@uerj.br † roditi@cbpf.br ‡ silvio.sorella@fis.uerj.br § arthurfvieira@if.ufrj.br arXiv:2504.13148v2 [hep-th] 30 Jun 2025 2 I. INTRODUCTION Quantum information theory has emerged as a powerful framework for analyzing fundamen...

arXiv 2025

[4] [4]

The field operators satisfy the canonical commutation relations: [ak,a † q]= 2π 2ωkδ(k−q), (A2) [ak,aq]= [a† k,a † q]= 0

Field expansion and commutation relations The massive scalar field φ(t,x) can be expressed in terms of plane waves as: φ(t,x)= ∫ dk 2π 1 2ωk (e−ikµxµ ak+eikµxµ a† k), (A1) 9 where ωk = k0 = √ k2+m2 is the relativistic energy dispersion relation. The field operators satisfy the canonical commutation relations: [ak,a † q]= 2π 2ωkδ(k−q), (A2) [ak,aq]= [a† k,...

work page

[5] [5]

These are defined as: ∆PJ (f,g )= ∫ d2xd2yf(x)∆PJ (x−y)g(y), H(f,g )= ∫ d2xd2yf(x)H(x−y)g(y)

Inner product and two-point functions With the smeared fields, the Lorentz-invariant inner product between two test functions f(x) and g(x) in the vacuum state is introduced by means of the two-point smeared Wightman function ⟨f∣g⟩= ⟨0∣φ(f)φ(g)∣0⟩= i 2∆PJ (f,g )+H(f,g ), (A6) where ∆PJ (f,g ) andH(f,g ) are the smeared versions of the Pauli-Jordan and Had...

work page

[6] [6]

Physical interpretation and causality Both the Hadamard and Pauli–Jordan distributions are Lorentz-invariant, but they fulfill distinct roles in the structure of quantum field theory. The Pauli–Jordan distribution, ∆ PJ (x), is particularly significant in encoding relativistic causality, as it vanishes for spacelike-separated arguments, thereby ensuring t...

work page

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M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information (Cambridge university press, 2010)

2010

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Faulkner, T

T. Faulkner, T. Hartman, M. Headrick, M. Rangamani, and B. Swingle, in Snowmass 2021 (2022) arXiv:2203.07117 [hep-th]

arXiv 2022