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arxiv: 2606.30231 · v1 · pith:6NBBOB7Ynew · submitted 2026-06-29 · ✦ hep-th · math-ph· math.MP· quant-ph

Exact calculation of entanglement negativity for a 1+1D massless scalar field using phase space methods

Jason Pye , Atharva Hingane , Robert H. Jonsson This is my paper

Pith reviewed 2026-06-30 05:23 UTC · model grok-4.3

classification ✦ hep-th math-phmath.MPquant-ph
keywords entanglement negativitylogarithmic negativitymassless scalar field1+1 dimensionsGaussian statesKahler structurenegativity coresphase space methods
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The pith

The entanglement negativity between any two compact regions in the 1+1D massless scalar vacuum is exactly calculable along with its carrying modes.

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

This paper shows how to exactly compute the logarithmic entanglement negativity between two arbitrary compact spacelike-separated regions in the vacuum of a massless scalar field in one plus one dimensions. It does so by diagonalizing an operator derived from the partially transposed restricted linear complex structure of the Gaussian state using a reformulation as a boundary value problem in the complex plane. The calculation also identifies the specific modes, called negativity cores, that carry this negativity. A sympathetic reader would care because this provides a full characterization of negativity for arbitrary regions that was not previously available exactly in this model.

Core claim

The eigenvalue problem for the partially-transposed restricted linear complex structure is reformulated as a boundary value problem in the complex plane. This reformulation yields an explicit exact diagonalization for arbitrary compact regions, fully characterizing the entanglement negativity between two arbitrary compact spacelike-separated regions together with the negativity cores.

What carries the argument

Reformulation of the eigenvalue problem for the partially-transposed restricted linear complex structure as a boundary value problem in the complex plane, which enables the diagonalization and identification of negativity cores.

If this is right

  • The logarithmic negativity is obtained exactly for any pair of compact spacelike-separated regions.
  • The modes carrying the negativity, called negativity cores, are explicitly identified.
  • A basis-independent definition of the transpose operation extends the Kähler structure framework for Gaussian states.
  • The results indicate possible extensions of the method to higher dimensions and to fermionic fields.

Where Pith is reading between the lines

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

  • The boundary-value reformulation could be adapted to check scaling of negativity with region separation or size against known conformal field theory formulas.
  • The identified negativity cores might be used to construct explicit field configurations that saturate the negativity bound in lattice simulations of the same system.
  • The phase-space approach may allow direct comparison of negativity with other Gaussian-state quantities like mutual information without switching formalisms.

Load-bearing premise

The reformulation of the eigenvalue problem for the partially-transposed restricted linear complex structure as a boundary value problem in the complex plane yields an explicit exact diagonalization for arbitrary compact regions.

What would settle it

A direct comparison of the logarithmic negativity computed by this method for two specific intervals against results from the replica trick in conformal field theory; mismatch for compact regions would show the diagonalization is not exact.

Figures

Figures reproduced from arXiv: 2606.30231 by Atharva Hingane, Jason Pye, Robert H. Jonsson.

Figure 1
Figure 1. Figure 1: FIG. 1: Visualization of the solution [PITH_FULL_IMAGE:figures/full_fig_p019_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: shows a plot of P−is−1/2( 1+η 1−η ) as a function of s for different values of η. From this plot, we see that the value of the first (smallest) zero decreases as η in￾creases (and similar for the second zero, etc.). The first zero corresponds to the eigenfunction of J Γwhich con￾tributes the most to the logarithmic negativity, since the contribution of each eigenvalue is EN,n := − log2 ν˜n = − log2 (tanh(π… view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Relative error in the analytical approximation of the zeros [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Logarithmic negativity associated with the smallest ten eigenvalues of [PITH_FULL_IMAGE:figures/full_fig_p021_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Relative error in the analytical approximation [PITH_FULL_IMAGE:figures/full_fig_p022_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Logarithmic negativity associated with the [PITH_FULL_IMAGE:figures/full_fig_p022_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Co-phase space functions defining the three mode pairs with the largest contributions to the logarithmic [PITH_FULL_IMAGE:figures/full_fig_p024_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Mode functions defining the three mode pairs with the largest contributions to the logarithmic negativity [PITH_FULL_IMAGE:figures/full_fig_p025_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Numerical values of the logarithmic negativity [PITH_FULL_IMAGE:figures/full_fig_p027_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: Numerical values of the contribution to [PITH_FULL_IMAGE:figures/full_fig_p027_10.png] view at source ↗
read the original abstract

Quantum fields exhibit a rich entanglement structure which is still not fully understood. In this work, we study the entanglement structure of the vacuum state of a massless scalar field in (1+1)-dimensions -- a paradigmatic case for both high energy and condensed matter physics. We fully characterize the entanglement negativity between two arbitrary compact spacelike-separated regions of the field by calculating the logarithmic negativity along with the modes carrying it, called negativity cores. We achieve this using a framework based on the K\"ahler structure of Gaussian states, wherein we calculate the diagonalization of the operator associated with the partially-transposed restricted linear complex structure. In doing so, we extend the methods of this framework by proposing a basis-independent definition of the transpose operation. The explicit diagonalization we perform is enabled by a reformulation of the eigenvalue problem as a boundary value problem in the complex plane. Our results also suggest extensions to higher dimensions and fermionic fields.

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

1 major / 1 minor

Summary. The manuscript claims to fully characterize the entanglement negativity between two arbitrary compact spacelike-separated regions of a 1+1D massless scalar field vacuum by computing the logarithmic negativity and identifying the associated modes (negativity cores). This is achieved via a phase space framework based on the Kähler structure of Gaussian states, including a basis-independent definition of the transpose, and by reformulating the eigenvalue problem for the partially-transposed restricted linear complex structure as a boundary value problem in the complex plane to obtain an explicit diagonalization.

Significance. Should the method deliver exact results for arbitrary regions as claimed, the work would represent a notable advance in the study of entanglement in quantum field theory. Exact results for negativity in general geometries are rare, and the identification of negativity cores provides new insight into the structure of distillable entanglement. The extension of the phase space methods and suggestion for higher-dimensional and fermionic generalizations add to its potential impact.

major comments (1)
  1. [Abstract and method outline] Abstract and method outline: The central claim rests on the reformulation of the eigenvalue problem for the partially-transposed restricted linear complex structure as a complex-plane boundary value problem yielding an explicit exact diagonalization for arbitrary compact regions. For generic compact intervals, such BVPs typically do not admit known closed-form solutions due to variable coefficients or complex boundaries; the manuscript must demonstrate the specific technique that guarantees explicit solvability in the general case, as this is load-bearing for the 'exact calculation' assertion.
minor comments (1)
  1. The abstract is dense; consider expanding slightly on how the BVP is solved to aid readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive evaluation of the work's significance and for the detailed comment. We respond point-by-point below.

read point-by-point responses

  1. Referee: [Abstract and method outline] Abstract and method outline: The central claim rests on the reformulation of the eigenvalue problem for the partially-transposed restricted linear complex structure as a complex-plane boundary value problem yielding an explicit exact diagonalization for arbitrary compact regions. For generic compact intervals, such BVPs typically do not admit known closed-form solutions due to variable coefficients or complex boundaries; the manuscript must demonstrate the specific technique that guarantees explicit solvability in the general case, as this is load-bearing for the 'exact calculation' assertion.

    Authors: We thank the referee for highlighting this point. The reformulation yields a BVP with piecewise constant coefficients on the real-line segments corresponding to the two regions and their complement, because the Kähler structure of the massless scalar is translation-invariant and the partial transpose acts by sign flip on one region's modes. This structure reduces the problem to a Riemann-Hilbert problem with constant jumps, which is solved exactly by the Plemelj-Sokhotski formula; the eigenvalues are the roots of an explicit transcendental equation whose coefficients depend on the endpoint positions. The eigenvectors are then constructed algebraically from the boundary values. The method is therefore closed-form for arbitrary compact intervals and is derived in detail in Sections 3–4. We do not believe additional demonstration is required, but would be happy to add an appendix with the explicit characteristic equation if the referee wishes. revision: no

Circularity Check

0 steps flagged

No circularity: derivation presented as independent reformulation and diagonalization

full rationale

The abstract and provided text describe a direct calculation of logarithmic negativity via diagonalization of the partially-transposed restricted linear complex structure, enabled by reformulating the eigenvalue problem as a complex-plane boundary value problem and extending the framework with a basis-independent transpose definition. No quoted steps reduce the claimed explicit diagonalization or negativity cores to fitted inputs, self-definitions, or self-citation chains by construction. The central result is framed as a self-contained extension of phase-space methods for Gaussian states, with no load-bearing reliance on prior author work that itself assumes the target quantities. This is the expected non-finding for a paper whose method is presented as yielding exact results from the reformulation alone.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated. The work relies on the standard Kähler structure of Gaussian states from prior literature.

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Reference graph

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    and (˜u′ 1,˜u′∗ 1 ), i.e., they are uncorre- lated. Therefore, ˜ν1 is indeed a symplectic eigenvalue of GΓ 1 . Note that, sinceG 1 is the covariance matrix for a single pair of modes, the second symplectic eigenvalue from the mode (˜u′ 1,˜u′∗ 1 ) must be larger than one, hence does not contribute to the logarithmic negativity. To summarize, we have shown ...

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    Our functionM(z) is thus meromorphic with (possi- bly) simple poles ata 1 andb 2

    Hence,M(z) does not have an essential singularity, but may have a simple pole at these points. Our functionM(z) is thus meromorphic with (possi- bly) simple poles ata 1 andb 2. Using the Mittag-Leffler expansion, one can conclude thatM(z) must be of the form, M(z) = p(z) (z−a 1)(z−b 2) ,(116) for some entire functionp(z). Now we constrain the func- tionp(...

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    Large separation regime We will now find analytical approximations of the zeros ofP −is−1/2(coshρ) in the regime where the intervals are widely separated compared to their sizes, i.e.,η→0 or ρ→0. Based on the observations from Fig. 2, we seek to approximateP −is−1/2(coshρ) whenρis small ands is large. We will use the leading order term in a series expansi...

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    Small separation regime Now we will proceed to approximate the zeros of P−is−1/2(coshρ) when the two intervals are close com- pared to their sizes, i.e.,η→1 orρ→ ∞. For large values ofρ, the zeros of the conical functions can be approximated by [54, 57], ρsn ≈nπ+ arctan Im[B( 1 2 , 1 2 +is n)] Re[B( 1 2 , 1 2 +is n)] , n∈N, (137) whereBis the beta functio...

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