A Study of Spin 1 Unruh–De Witt Detectors

: A study of the interaction of spin 1 Unruh–De Witt detectors with a relativistic scalar quantum field is presented here. 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 decrease in the size of violation, in the case of spin 1, both a decrease or an increase in the size of the violation may occur. This effect is ascribed to the fact that Tsirelson’s bound is not saturated in the case of qutrits.

In this work, we shall employ spin 1 Unruh-De Witt detectors to investigate the effects of a quantum relativistic scalar field on the Bell-CHSH inequality [8,9], following the setup already outlined for spin 1/2 detectors [10].More precisely, we consider the interaction of a pair of qutrits with a real Klein-Gordon field in Minkowski spacetime, by taking as initial field configuration the vacuum state |0⟩.
One starts with the density matrix corresponding to the state where is the maximally entangled singlet state of two qutrits.As it is customary, (A, B) refer to Alice and Bob, respectively.
One let evolve the density matrix by means of the unitary operator corresponding to the dephasing channel.The resulting asymptotic density matrix is employed to study the effects arising from the presence of the scalar field on the violations of the Bell-CHSH inequality.Though, one has to remember that the operators (A, A ′ ) and (B, B ′ ) entering the Bell-CHSH correlator C, Eq. (21), are required to fulfill the conditions (22), implying that (A, A ′ ) and (B, B ′ ) have to be space-like, in agreement with relativistic causality.This feature is taken into account by employing the right and left wedges, W R , W L : Regions contained in W R are space-like with respect to regions of W L .
Moreover, as one learns from [11,12], the use of the wedges regions (W R , W L ) enables us to employ the results about the nature of the vacuum state |0⟩.It has been established [11,12] that the vacuum |0⟩ is highly entangled, exhibiting maximal violation of the Quantum Field Theory formulation of the Bell-CHSH inequality for regions belonging to the wedges.As such, the state (1) looks ideal for a study of the effects of the quantum field on the violation of the Bell-CHSH inequality for the qutrits system.
Nevertheless, as we shall see, there are remarkable differences between the spin 1/2 and the spin 1 cases.As far as the Bell-CHSH inequality is concerned, for spin 1/2, the effects induced by the quantum field result in a decreasing of the size of the violation, due to the fact that the Tsirelson bound [13], i.e. 2 √ 2, is already attained in the absence of the field.As the Tsirelson bound is the maximum allowed value for the violation, one can easily figure out that the presence of a quantum field can only induce a decreasing of the size of the violation, see [10] for more details.Instead, in the case of spin 1, the situation looks rather different.Here, it is known that Tsirelson's bound is never attained [14,15].The maximum value for the Bell-CHSH inequality is approximately 2.55.As such, depending on the choice of the parameters, the effects of the quantum field may give rise either to a decreasing or to an increasing of the violation, while remaining compatible with Tsirelson's bound 2

√
2. In the case of a decreasing one has a degradation [16] of the entanglement properties of the initial state, while in the case of an increasing of the violation one might speak of extraction of entanglement [1][2][3][4][5][6][7].
This work is organized as follows.In Sec.(II), we evaluate the qutrit density matrix by considering the dephasing coupling regime.In Sec.(III) we provide an overview of the fundamental characteristics of the Weyl operators and their von Neumann algebra, introducing key concepts that will be employed throughout this study.In Sec.(IV), we discuss how the effects of the quantum field φ on the violation of the Bell-CHSH inequality, which can be obtained in closed form by using the powerful modular theory of Tomita-Takesaki [11,12,[18][19][20].Notably, it turns out that the violation of the Bell-CHSH inequality exhibits both an increasing and a decreasing behavior as compared to the case in which the field φ is absent, Sec.(V) collects our conclusion.

A. Preliminaries
For the initial density matrix we have where The time evolution of ρ ABφ (0) is governed by the unitary operator where the operator J z corresponds to the component of spin along the z-axis, and φ(f j ), j = A, B, is the smeared field [17]: where f j (x) are smooth test functions with compact support1 , f j (x) ∈ C ∞ 0 (R 4 ).As mentioned before, the support of Alice's test function f A (x) is an open region O ∈ W R .Relying thus on the powerful Tomita-Takesaki modular theory for von Neumann algebras [11,12,[18][19][20], Bob's test function f B (x) will be supported in the causal complement O ′ of O, located in W L .The norms and the Lorentz invariant inner products of (f A , f B ) are also determined by the properties of the modular theory, as given in Eqs.(62), see the review [21] for a detailed account.The role of the test functions f j is that of localizing the quantum field in the regions mentioned above.
For the quantum field φ, one writes Let us proceed by providing the derivation of the unitary evolution operator of Eq.( 7).One starts with the Hamiltonian where H 0 stands for the free Hamiltonian and H I (t) is the interaction term: Notice that For the evolution operator in the interaction representation, we have where T t is the time ordering.In order to work out expression (14), one makes use of the Magnus formula [2,3], summarized as with ] We remind now that the field commutator [φ(x), φ(y)] is a c-number.As a consequence, Ω j = 0, i ≥ 3, while Ω 2 yields an irrelevant phase.Therefore, up to an irrelevant phase, for the evolution operator U(t), one gets Therefore, in the large asymptotic time, t → ∞, equation ( 7) follows, namely, at large time, the density matrix is written as The subsequent stage involves deriving the density matrix ρAB for the qutrit system through the process of tracing out the field modes: Finally, once the density matrix ρAB is known, one is capable of evaluating the Bell-CHSH correlator where with (A, A ′ ), (B, B ′ ) being the Bell operators, namely Concerning the commutators [A, A ′ ] and [B, B ′ ], they can be expressed in terms of the four Bell's parameters (α, α ′ , β, β ′ ), Eq.( 63), i.e. and The Bell-CHSH inequality is said to be violated whenever

II. EVALUATION OF THE QUTRIT DENSITY MATRIX IN THE CASE OF THE DEPHASING COUPLING DETECTORS
We shall consider the density matrix ρAB in he so-called dephasing coupling regime [2,3], for which the evolution operator is given by U = U A ⊗ U B , where the unitary operator for the detector j = A, B is with the commutation relation The above commutation relation follows from the fact that Alice's and Bob's test functions (f A , f B ) are space-like.This feature enables for several practical simplifications in the evaluation of the resulting density matrix for the qutrits system.
Using the algebra of the spin 1 matrices, the expression (26) can be written as where c j ≡ cos φ(f j ) and s j ≡ sin φ(f j ).With the initial density matrix ρ ABφ (0) given in Eq.( 5), its evolution is described as follows: Tracing over φ, we get a rather lengthy expression for ρAB , namely where ⟨s A s B (c A − 1)(c B − 1)⟩, etc., denotes the expectation value of where In the next section we shall see how can these correlation functions be addressed in closed form, using the Tomita-Takesaki theory.

III. TOMITA-TAKESAKI MODULAR THEORY AND THE VON NEUMANN ALGEBRA OF THE WEYL OPERATORS
To calculate the correlation functions of the Weyl operators, Eq.(31), it is worth providing a compact review of the properties of the von Neumann algebra related to such operators.For a more detailed review, one can check Refs.[20,21].
Let us begin by recalling the commutator between the scalar fields, for arbitrary spacetime separation where the Lorentz-invariant causal Pauli-Jordan distribution ∆ P J (x − y) is defined by with ε(x) ≡ θ(x) − θ(−x).The Pauli-Jordan distribution ∆ P J (x − y) vanishes outside of the light cone, guaranteeing that measurements at points separated by space-like intervals do not interfere, that is Now, let O be a subregion of the W R and let M(O) be the space of smooth test functions with support contained in O, namely Following [11,12] one introduces the symplectic complement of M(O) as This symplectic complement M ′ (O) comprises all test functions for which the smeared Pauli-Jordan expression ∆ P J (f, g) vanishes for any f belonging to allowing us to rephrase causality, Eq.( 35), as [11,12] [φ(f ), φ(g)] = 0, (39) whenever f ∈ M(O) and g ∈ M ′ (O).
As already mentioned in Sec.(I), the so-called Weyl operators [11,12,20] play an important role in the study of the Bell-CHSH inequality.This class of unitary operators is obtained by exponentiating the smeared field By applying the Baker-Campbell-Hausdorff formula together with the commutation relation (34), one finds that the Weyl operators lead to the following algebraic structure: Moreover, if f and g are space-like, the Weyl operators W f and W g commute.By expanding the field φ in terms of creation and annihilation operators, one can evaluate the expectation value of the Weyl operator, finding where ||h|| 2 = ⟨h|h⟩ and is the Lorentz invariant inner product between the test functions (f, g) [11,12,20], with the usual relation A von Neumann algebra A(M) arises by taking all possible products and linear combinations of the Weyl operators defined on M(O).In particular, the Reeh-Schlieder theorem [11,12,17,18], states that the vacuum state |0⟩ is both cyclic and separating for the von Neumann algebra A. Consequently, we can apply the Tomita-Takesaki modular theory [11,12,[18][19][20] and introduce the anti-linear unbounded operator S, whose action on the von Neumann algebra A(M) is defined as from which it follows that S 2 = 1 and S|0⟩ = |0⟩.The operator S has a unique polar decomposition [19]: where J is anti-unitary and ∆ is positive and self-adjoint.These operators are characterized by the following set of properties [11,12,[18][19][20]: From the Tomita-Takesaki theorem [11,12,[18][19][20], it follows that JA(M)J = A ′ (M), meaning that, upon conjugation by the operator J, the algebra A(M) is mapped onto its commutant A ′ (M), namely: Furthermore, the theorem states that there is a one-parameter family of operators ∆ it , t ∈ R, that leave the algebra A(M) invariant, such that the the following equation holds The Tomita-Takesaki modular theory is particularly well-suited for analyzing the Bell-CHSH inequality within the framework of relativistic Quantum Field Theory [11,12].As demonstrated in [20], it provides a purely algebraic method for constructing Bob's operators from Alice's ones by using the modular conjugation J. Given Alice's operator A f , one can assign to Bob the operator B f = JA f J, ensuring their mutual commutativity due to the Tomita-Takesaki theorem, as B f = JA f J belongs to the commutant A ′ (M) [20].
An important outcome of the Tomita-Takesaki modular theory, established by [22,23], allows the extension of the action of the modular operators (J, ∆) to the space of the test functions.In fact, when equipped with the Lorentz-invariant inner product ⟨f |g⟩, Eq.( 43), the set of test functions forms a complex Hilbert space F that possesses a variety of properties.To be more precise, it is found that the subspaces M and iM are standard subspaces for F [22].This implies that: As shown in [22], for such subspaces, it's viable to establish a modular theory similar to that of the Tomita-Takesaki theory.This involves introducing an operator s acting on M + iM such that for f, h ∈ M. With this definition, it's worth noting that s 2 = 1.Employing the polar decomposition, one obtains: where j is an anti-unitary operator, while δ is positive and self-adjoint.Similarly to the operators (J, ∆), the operators (j, δ) fulfill the following properties [22]: Further, one can show [12,22] that a test function f belongs to M if and only if Indeed, let us suppose that f ∈ M. From Eq.(48), one can express for some (h 1 , h 2 ).Since s 2 = 1 it follows that so that h 1 = f and h 2 = 0. Similarly, one has that Thus, the lifting of the action of the operators (J, ∆) to the space of test functions is accomplished by [23] Je iφ(f ) J = e −iφ(jf ) , ∆e iφ(f ) ∆ −1 = e iφ(δf ) .
Also, it is important to note that if f ∈ M ⇒ jf ∈ M ′ .This property follows from It is worth reminding here that δ is an unbounded operator with continuous spectrum.For instance, as one learns from the work of [24], for the wedge W R , the spectrum of δ coincides with the positive real line, i.e., log(δ) = R.In the case of a continuous spectrum we lack the notion of eigenstates.Rather, it is appropriate to make use of the spectral decomposition [19] of the operator δ and refer to spectral subspaces σ λ , parametrized by a real parameter λ ∈ R + .
We have now all the necessary ingredients to evaluate the correlation functions of the Weyl operators.By examining expression (30), one recognizes that the fundamental quantity to be computed is of the form so that we need to evaluate the following norms (||f A || 2 , ||f B || 2 ) and the inner product ⟨f A |f B ⟩.We begin by focusing on Alice's test function f A .We require that f A ∈ M(O) where O is located in the right Rindler wedge.Following [11,12,20], the test function f A can be further specified by considering the spectrum of the operator δ.By selecting the subspace σ λ = [λ 2 − ε, λ 2 + ε] and introducing the normalized vector ϕ belonging to this subspace, one writes where η is an arbitrary parameter.As required by the setup outlined above, Eq.(57) ensures that We observe that jϕ is orthogonal to ϕ, i.e., ⟨ϕ|jϕ⟩ = 0.In fact, from it follows that the modular conjugation j exchanges the spectral subspace Regarding Bob's test function f B , we use the modular conjugation operator j and define ensuring that This implies that, as required by the relativistic causality, f B belongs to the symplectic complement M ′ (O), located in the left Rindler wedge, namely: f B ∈ M ′ (O).Finally, considering that ϕ belongs to the spectral subspace [λ 2 −ε, λ 2 +ε], it follows that [20], which provide us the needed inner products.
Reminding that the initial state for AB is • The contributions arising from the scalar field φ are encoded in the exponential terms e −4η2 (1−λ) 2 and e −2η 2 (1+λ 2 ) .It is worth reminding here that the parameter η 2 is related to the norm of the test function f A , Eqs.(62), that is, this parameter reflects the freedom one has in defining the test function f A through the operator s.As pointed out in [20,25], η is a free parameter appearing in the Quantum Field Theory formulation of the Bell-CHSH inequality in terms of Weyl operators, playing a similar role of the free Bell's angles and it can bee chosen in the most suitable way.This feature can be understood as follows.Looking at the Bell's operators (A, A ′ , B, B ′ ), Eqs.(63), one realizes that they are dichotomic for arbitrary values of the parameters (α, α ′ , β, β ′ ).As such, they are completely free and, in fact, are chosen at the best convenience in the final expression of the Bell-CHSH inequality.The same pattern is encountered in the case of the parameter η.One has to notice that the Weyl operator = e iφ(η(1+s)ϕ) , is unitary for any value of the parameter η.
• We have now to face the choice of the spectral subspace σ λ of the modular operator δ.This is a not easy task due to the fact that δ has a continuous spectrum given by the positive real line R + .For a better illustration of this point, we remind here the expression found in [12] for the violation of the Bell-CHSH in the vacuum state of a quantum scalar field 2 , namely Let us end this section by reminding the results obtained in the case of spin 1/2, in the dephasing channel, as reported in [10].
For the initial state we have The Bell-CHSH inequality is found which, unlike the case of the spin 1, exhibits only a decreasing of the size of the violation.