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I. Roditi

Publications and source records attributed to I. Roditi.

At least 19 recordsLinked to original sources
arXiv2026

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

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.

quant-ph↗
arXiv2026

Majorana bilinears, compact modular seeds and the approach to the Tsirelson bound

In this work we study the Bell-CHSH inequality in the vacuum of the free massive Majorana field in $1+1$ 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 $i$, 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 $λ=1$ of the modular operator. This establishes in a precise way how the even sector of the free Majorana field realizes the Summers-Werner mechanism.

hep-th↗
arXiv2026

Mass Dependence of Araki Relative Entropy through Modular Theory

Building on the established one-particle formula for the Araki relative entropy of coherent states, we study how its value acquires a nontrivial dependence on the mass of the scalar field. For a localized vector $h$ belonging to the standard subspace $H_m$ of the one-particle Hilbert space, the known quadratic-form expression is: $S_{H_m}(h)=-\langle h,\logδ_{H_m}\,h\rangle$. Our contribution is to construct explicitly a mass-indexed family of vectors of the wedge standard subspace on which this expression is evaluated. The mass-shell map $h_m=E_mf$ organizes four structural conditions on rapidity representatives---on-shell dependence, a controlled massless boundary value, decay for large real rapidity, and Bisognano--Wichmann strip analyticity---and we exhibit an entire rapidity wave function, built from a doubled light-cone phase, two sinc factors, and a Gaussian pair, that satisfies them together with the sharp localization criterion: Hardy-type $L^2$ control throughout the Bisognano--Wichmann strip and the exact Tomita boundary relation. The family therefore belongs to $H_m(\W_R)$ for every $m>0$, and its Araki relative entropy is finite and strictly positive, with an exact spectral representation that makes positivity manifest. The entropy is strongly suppressed at large mass, attains a maximum at intermediate mass in $1+1$ dimension, and converges to a finite value along the modular flow as $m\to0^+$. The construction extends fiberwise to $1{+}d$ dimensions through the transverse mass.

hep-th↗
arXiv2026

Finite Weyl polynomials and the approach to Tsirelson's bound in relativistic scalar quantum field theory

We construct four bounded Hermitian operators, each one a finite polynomial in the unitary Weyl operators, for the Bell-CHSH inequality in a free massive scalar field in $1+1$ dimensions. The operators are localized in complementary wedges. Odd Weyl harmonics provide two exactly anticommuting axis observables in each wedge, while normalizable packets with compact support in the spectrum of the boost generator give exact modular inner products at nonvanishing bandwidth. The resulting Bell-CHSH correlator is a finite double sum. An operator with six Weyl terms per axis already gives $2.14885$. Using normalized Fejér approximants of $\operatorname{sgn}(\cos(x))$, we show that the supremum over the finite-polynomial family equals $2\sqrt{2}$, although no finite member attains it; a degree-$511$ example gives $2.80027$. We also point out that, in a centered quasifree state, a Bell-CHSH test whose four final settings are bounded functions of individual quadratures admits one common Gaussian representation and therefore stays below $2$. The finite-Weyl construction avoids this restriction because Bob's final settings mix two noncommuting axis observables.

hep-th↗
arXiv2026

Modular wedge localization, Majorana fields and the Tsirelson limit of the Bell-CHSH inequality

The massive Majorana field in $1+1$ 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 $h^2(ω)$ for the modular operator. The resulting analytic families approach the Tsirelson bound in the vacuum state as their spectral weight concentrates near $ω\approx0$, corresponding to the eigenvalue $λ^2 \approx 1$ of the modular operator.

hep-th↗
arXiv2026

Modular Theory and the Bell-CHSH inequality in relativistic scalar Quantum Field Theory

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 $1+1$ 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.

hep-th↗
arXiv2026

Cat states and violation of the Bell-CHSH inequality in relativistic Quantum Field Theory

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 $sign(φ(f))$, where $φ(f)$ 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.

hep-th↗
arXiv2026

Bell-CHSH inequality and unitary transformations in Quantum Field Theory

Unitary transformations are employed to enhance the violations of the Bell-CHSH inequality in relativistic Quantum Field Theory. The case of the scalar field in $1+1$ Minkowski space-time is scrutinized by relying on the Tomita-Takesaki modular theory. The example of the bounded Hermitian operator $sign(φ(f))$, where $φ(f)$ stands for the smeared scalar field, is worked out. It is shown that unitary deformations enable for violations of the Bell-CHSH inequality. The setup is generalized to the Proca vector field by means of its equivalence with the scalar theory.

quant-ph↗
arXiv2025

On a class of bounded Hermitian operators for the Bell-CHSH inequality in Quantum Field Theory

The violation of the Bell-CHSH inequality in a relativistic scalar Quantum Field Theory is analysed by means of a set of bounded Hermitian operators constructed out of the unitary Weyl operators. These operators allow for both analytic and numerical approaches. While the former relies on the modular theory of Tomita-Takesaki, the latter is devised through an explicit construction of the test functions needed for the localization of the aforementioned operators. The case of causal tangent diamonds in $1+1$ Minkowski spacetime is scrutinized.

quant-ph↗
arXiv2025

Bell and Mermin inequalities in Quantum Field Theory from vacuum projectors and Weyl operators

The use of the vacuum projector $|0 \rangle \langle 0| $ and of the unitary Weyl operators enables us to construct a set of Hermitian dichotomic operators in relativistic scalar Quantum Field Theory in Minkowski spacetime. Employing test functions supported in diamond regions, both Bell and Mermin inequalities are studied by means of a numerical setup. In addition to reporting expressive violations of both inequalities, the cluster property is also checked.

quant-ph↗
arXiv2025

Bell's inequality in relativistic Quantum Field Theory

A concise and self-contained introduction to the Bell inequality in relativistic Quantum Field Theory is presented. Taking the example of a real scalar massive field, the violation of the Bell inequality in the vacuum state and for causal complementary wedges is illustrated.

quant-ph↗
arXiv2024

Introduction to Bell's inequality in Quantum Mechanics

A pedagogical introduction to Bell's inequality in Quantum Mechanics is presented. Several examples, ranging from spin $1/2$ to coherent and squeezed states are worked out. The generalization to Mermin's inequalities and to GHZ states is also outlined.

quant-ph↗
arXiv2024

Investigation of the Bell-CHSH inequality in diamond regions

A numerical setup for the Bell-CHSH inequality for causal diamonds in $1+1$ Minkowski spacetime is presented. Upon choosing a suitable set of test functions supported in the diamonds, sensible violations are reported for the correlation function of Weyl operators of a real scalar massive field in the vacuum state.

quant-ph↗
arXiv2024

Bell-CHSH inequality and unitary operators

Unitary operators are employed to investigate the violation of the Bell-CHSH inequality. The ensuing modifications affecting both classical and quantum bounds are elucidated. The relevance of a particular class of unitary operators whose expectation values are real is pointed out. For these operators, the classical and quantum bounds remain unaltered, being given, respectively, by $2$ and $2\sqrt{2}$. As an example, the Weyl unitary operators for a real scalar field in relativistic Quantum Field Theory are discussed.

quant-ph↗
arXiv2024

Gluing together Quantum Field Theory and Quantum Mechanics: a look at the Bell-CHSH inequality

The Bell-CHSH inequality in the vacuum state of a relativistic scalar quantum field is revisited by making use of the Hilbert space ${\cal H} \otimes {\cal H}_{AB}$, where ${\cal H}$ and ${\cal H}_{AB}$ stand, respectively, for the Hilbert space of the scalar field and of a generic bipartite quantum mechanical system. The construction of Hermitian, field-dependent, dichotomic operators is devised as well as the Bell-CHSH inequality. Working out the $AB$ part of the inequality, the resulting Bell-CHSH correlation function for the quantum field naturally emerges from unitary Weyl operators. Furthermore, introducing a Jaynes-Cummings type Hamiltonian accounting for the interaction between the scalar field and a pair of qubits, the quantum corrections to the Bell-CHSH inequality in the vacuum state of the scalar field are evaluated till the second order in perturbation theory.

quant-ph↗
arXiv2024

A study of the spin 1 Unruh-De Witt detectors

A study of the spin 1 Unruh-De Witt detectors interacting with a relativistic scalar quantum field is presented. After tracing out the field modes, the resulting density matrix for a bipartite qutrit system is employed to investigate the violation of the Bell-CHSH inequality. Unlike the case of spin $1/2$, for which the effects of the quantum field result in a decreasing of the size of violation, in the case of spin $1$ both decreasing and increasing of the violation may occur. This effect is ascribed to the fact that Tsirelson's bound is not saturated in the case of qutrits.

hep-th↗