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arXiv:2610.33773

Induced Bell-CHSH observables in the Fewster-Verch measurement scheme

M. S. Guimaraes, I. Roditi, S. P. Sorella

quant-phhep-thmath-phSubmitted Oct 24, 2026• Announced Oct 27, 2026• 0 citations
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Abstract

The Summers-Werner theorem shows that the vacuum of a relativistic quantum field theory violates the Bell-CHSH inequality maximally on complementary wedges, but it is an existence statement: it exhibits neither the observables nor an apparatus that measures them. We give an explicit operator-level realization within the Fewster-Verch measurement framework, obtaining dichotomic Bell observables of a free Majorana field in $1+1$ dimensions as observables induced by local couplings to Majorana probes. The induced system smearing factorizes exactly as $f_h=k\,b_{λ,h}$, so its support is controlled by the compactly supported coupling profile $k$; dichotomy is imposed by calibrating and normalizing the induced smearings rather than required of the probe observable; and a probe-only normal ordering yields the product of Alice's and Bob's induced bilinears, which the non-multiplicative induced map alone does not provide. Under explicit attainability hypotheses the induced observables saturate the Tsirelson bound in a state prepared from the vacuum. The construction extends to coupled scalar fields.

AI Summary

The paper asks how Bell–CHSH observables can arise from explicitly localized system–probe interactions in the Fewster–Verch (FV) measurement framework, rather than from algebraic existence arguments. It studies a free massive Majorana field in $1+1$ dimensions coupled to Majorana probes through compactly supported mass mixing. Assuming a well-defined, resumable Møller construction, the induced system smearing factorizes as $f_h=k b_{\lambda,h}$, giving a representative supported inside the coupling region. Given independent non-null modes in the attainable output range, real Gram–Schmidt calibration and CAR normalization turn contraction-corrected probe bilinears into Hermitian system involutions. Expanding joint probe expressions and subtracting probe-only contractions yields products of the induced Alice–Bob bilinears without assuming multiplicativity of the FV map. With three attainable real modes per party in spacelike separated regions, the construction generates two commuting even Pauli algebras. A joint singlet projection, preceded if necessary by a local Pauli rotation, prepares a state from the vacuum in which suitable settings attain $2\sqrt2$. This is not maximal violation in the original vacuum. A scalar-field extension establishes finite-harmonic Weyl identities and an induced sign–cosine limit. The attainable ranges are assumed rather than characterized. Induced involutions generally reproduce only first moments, not sharp binary probe measurements; the joint preparation is not implemented by the proposed FV scheme. In the scalar extension, positive probe noise prevents the infinite corrected Fourier prescription from defining a bounded readout. A complete operational Bell protocol remains unresolved.

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The Summers-Werner theorem shows that the vacuum of a relativistic quantum field theory violates the Bell-CHSH inequality maximally on complementary wedges, but it is an existence statement: it exhibits neither the observables nor an apparatus that measures them. We give an explicit operator-level realization within the Fewster-Verch measurement framework, obtaining dichotomic Bell observables of a free Majorana field in 1+11+11+1 dimensions as observables induced by local couplings to Majorana probes. The induced system smearing factorizes exactly as fh=k bλ,hf_h=k\,b_{λ,h}fh​=kbλ,h​, so its support is controlled by the compactly supported coupling profile kkk; dichotomy is imposed by calibrating and normalizing the induced smearings rather than required of the probe observable; and a probe-only normal ordering yields the product of Alice's and Bob's induced bilinears, which the non-multiplicative induced map alone does not provide. Under explicit attainability hypotheses the induced observables saturate the Tsirelson bound in a state prepared from the vacuum. The construction extends to coupled scalar fields.
The paper asks how Bell–CHSH observables can arise from explicitly localized system–probe interactions in the Fewster–Verch (FV) measurement framework, rather than from algebraic existence arguments. It studies a free massive Majorana field in 1+11+11+1 dimensions coupled to Majorana probes through compactly supported mass mixing.
Assuming a well-defined, resumable Møller construction, the induced system smearing factorizes as fh=kbλ,hf_h=k b_{\lambda,h}fh​=kbλ,h​, giving a representative supported inside the coupling region. Given independent non-null modes in the attainable output range, real Gram–Schmidt calibration and CAR normalization turn contraction-corrected probe bilinears into Hermitian system involutions. Expanding joint probe expressions and subtracting probe-only contractions yields products of the induced Alice–Bob bilinears without assuming multiplicativity of the FV map.
With three attainable real modes per party in spacelike separated regions, the construction generates two commuting even Pauli algebras. A joint singlet projection, preceded if necessary by a local Pauli rotation, prepares a state from the vacuum in which suitable settings attain 222\sqrt222​. This is not maximal violation in the original vacuum. A scalar-field extension establishes finite-harmonic Weyl identities and an induced sign–cosine limit.
The attainable ranges are assumed rather than characterized. Induced involutions generally reproduce only first moments, not sharp binary probe measurements; the joint preparation is not implemented by the proposed FV scheme. In the scalar extension, positive probe noise prevents the infinite corrected Fourier prescription from defining a bounded readout. A complete operational Bell protocol remains unresolved.
In this work we study the Bell-CHSH inequality in the vacuum of the free massive Majorana field in 1+11+11+1 dimensions, taking as the dichotomic observables of the two parties even bilinears of smeared fields localized in the right and in the left Rindler wedges. The test functions are generated from seeds in the modular-frequency variable through the Tomita-Takesaki operator of the right wedge and, on the side of the left wedge, through its adjoint multiplied by iii, which accounts for twisted duality. For seeds supported at positive modular frequency the one-particle overlaps between the two parties do not depend on the modular weights, and we obtain the Bell parameter in closed form. When the seeds are phase rotations of a single compact bump, the bilinears generate a copy of the Pauli algebra in each wedge, and we compute exactly the correlation matrix of the vacuum restricted to the two copies. The optimal settings violate the classical bound up to spectral widths more than twice as large as those reached with planar settings. We also show that the Tsirelson bound is never attained in the wedge vacuum for any choice of dichotomic observables, and that approaching it forces the modular spectral weight of the Bell vectors to concentrate at the spectral point λ=1λ=1λ=1 of the modular operator. This establishes in a precise way how the even sector of the free Majorana field realizes the Summers-Werner mechanism.
The massive Majorana field in 1+11+11+1 dimension is employed to investigate the violation of the Bell-CHSH inequality in relativistic Quantum Field Theory. We give an explicit rapidity-space realization of the Summers-Werner modular-localization construction and reduce the vacuum Bell-CHSH correlator to a single spectral weight h2(ω)h^2(ω)h2(ω) for the modular operator. The resulting analytic families approach the Tsirelson bound in the vacuum state as their spectral weight concentrates near ω≈0ω\approx0ω≈0, corresponding to the eigenvalue λ2≈1λ^2 \approx 1λ2≈1 of the modular operator.
The Tomita-Takesaki modular theory is employed to discuss the Bell-CHSH inequality in wedge regions. By using the Bisognano-Wichmann results, the construction of a set of wedge localized vectors in the one-particle Hilbert space of a relativistic massive scalar field in 1+11+11+1 dimensions is devised to establish whether violations of the Bell-CHSH inequality might occur for different choices of Bell's operators. In particular, the construction of the wedge localized vectors employed in the seminal work by Summers-Werner is scrutinized and applied to Weyl and other operators. We also outline a possible path towards the saturation of Tsirelson's bound.
The Bell-CHSH inequality provides a powerful probe of quantum correlations in relativistic quantum field theory. We study its violation in the vacuum of the free massless Majorana field in 1+1 dimensions using modular wedge localization. By introducing a rapidity variable, we obtain a particularly simple representation of the one-particle Hilbert space as a direct sum of two scalar components, in which the spinor mixing present in previous formulations is eliminated. This allows us to give an explicit algorithm for constructing four bounded, Hermitian, dichotomic operators whose Bell-CHSH expectation value can be made arbitrarily close to the Tsirelson bound ( 2 2 ). We derive the resulting Bell-CHSH functional in terms of the modular spectral weight and provide a semi-analytic family of test functions that realizes this near-saturation explicitly.
A cat state localized in the right Rindler wedge is employed to study the violation of the Bell-CHSH inequality in a relativistic scalar free Quantum Field Theory. By means of the bounded Hermitian operator sign(φ(f))sign(\varphi(f))sign(φ(f)), where φ(f)\varphi(f)φ(f) stands for the smeared scalar field, it turns out that the Bell-CHSH correlator can be evaluated in closed analytic form in terms of the imaginary error function. Being the superposition of two coherent states, cat states allow for the existence of interference terms which give rise to a violation of the Bell-CHSH inequality. As such, the present setup can be considered as an explicit realization of the results obtained by Summers-Werner.