A Study of Spin 1 Unruh–De Witt Detectors

Abstract

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.

1. Introduction

The Unruh–De Witt (UDW) detectors are useful models which are broadly employed in the study of relativistic quantum information, as seen in [1,2,3,4,5,6,7].

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 the vacuum state $|0\rangle$ as the initial field configuration.

One starts with the density matrix corresponding to the state

$${|\psi \rangle }_{AB}\otimes |0\rangle ,$$

(1)

where

$$\begin{array}{c}\hfill {|\psi \rangle }_{AB}={\displaystyle \frac{1}{\sqrt{3}}}\left({|1\rangle }_A{\otimes |-1\rangle }_B-{|0\rangle }_A\otimes {|0\rangle }_B{+|-1\rangle }_A\otimes {|1\rangle }_B\right),\end{array}$$

(2)

is the maximally entangled singlet state of two qutrits. As is customary, $(A,B)$ refer to Alice and Bob, respectively.

One should then let the density matrix evolve by means of the unitary operator corresponding to the dephasing channel. The resulting asymptotic density matrix can be employed to study the effects on the violations of the Bell–CHSH inequality arising from the presence of the scalar field. However, one has to remember that the operators $(A,A^{\prime })$ and $(B,B^{\prime })$ entering the Bell–CHSH correlator $\mathcal{C}$ in Equation (21) are required to fulfill the conditions of Equation (22), implying that $(A,A^{\prime })$ and $(B,B^{\prime })$ have to be space-like in agreement with relativistic causality. This feature is taken into account by employing the right and left wedges, ${\mathcal{W}}^R$ and ${\mathcal{W}}^L$:

$${\mathcal{W}}^R=\{x;x>|t\left|\right\},{\mathcal{W}}^L=\{x;,-x>|t\left|\right\}.$$

(3)

Regions contained in ${\mathcal{W}}^R$ are space-like with respect to regions of ${\mathcal{W}}^L$.

Moreover, as one learns from [11,12], the use of the wedges’ regions $({\mathcal{W}}^R$ and ${\mathcal{W}}^L)$ enables us to employ the results about the nature of the vacuum state $|0\rangle$. It has been established [11,12] that the vacuum $|0\rangle$ 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, state (1) appears to be 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., $2\sqrt{2}$, 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 decrease in 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 decrease or to an increase in the violation, while remaining compatible with Tsirelson’s bound $2\sqrt{2}$. In the case of a decrease, there is a degradation [16] of the entanglement properties of the initial state, while in the case of an increase in the violation, one might speak of an extraction of the entanglement [1,2,3,4,5,6,7].

This work is organized as follows. In Section 2, we evaluate the qutrit density matrix by considering the dephasing coupling regime. In Section 3 we provide an overview of the fundamental characteristics of the Weyl operators $W_{f_j}$

$$W_{f_j}=e^{i\phi \left(f_j\right)},j=A,B,$$

(4)

and their von Neumann algebra, introducing key concepts that will be employed throughout this study. In Section 4, we discuss the effects of the quantum field $\phi$ on the violation of the Bell–CHSH inequality, which can be obtained in its closed form by using the powerful modular theory of Tomita–Takesaki [11,12,17,18,19]. Notably, it turns out that the violation of the Bell–CHSH inequality exhibits both an increasing and a decreasing behavior when compared to a case in which the field $\phi$ is absent. Section 5 presents our conclusion.

Preliminaries

For the initial density matrix we have

$$\begin{array}{c}\hfill {\rho }_{AB\phi }\left(0\right)={\rho }_{AB}\left(0\right)\otimes |0\rangle \langle 0|,\end{array}$$

(5)

where

$${\rho }_{AB}\left(0\right)={|\psi \rangle }_{ABAB}\langle \psi |.$$

(6)

The time evolution of ${\rho }_{AB\phi }\left(0\right)$ is governed by the unitary operator

$$\begin{array}{c}\hfill \mathcal{U}=e^{-i[J_A^z\otimes \phi \left(f_A\right)+J_B^z\otimes \phi \left(f_B\right)]},\end{array}$$

(7)

where the operator $J^z$ corresponds to the component of spin along the z-axis, and $\phi \left(f_j\right)$, $j=A,B$ is the smeared field [20]:

$$\phi \left(f_j\right)=\int d^{4}x\phi \left(x\right)f_j\left(x\right),j=A,B,$$

(8)

where $f_j\left(x\right)$ represents smooth test functions with compact support1, and $f_j\left(x\right)\in {\mathcal{C}}_0^{\infty }\left({\mathbb{R}}^4\right)$. As mentioned before, the support of Alice’s test function $f_A\left(x\right)$ is an open region $\mathcal{O}\in {\mathcal{W}}^R$. Relying thus on the powerful Tomita–Takesaki modular theory for von Neumann algebras [11,12,17,18,19], Bob’s test function $f_B\left(x\right)$ will be supported by the causal complement ${\mathcal{O}}^{\prime }$ of $\mathcal{O}$, located in ${\mathcal{W}}^L$. 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 Equation (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 $\phi$, one writes

$$\begin{array}{ccc}\hfill \phi \left(x\right)& =& \int {\displaystyle \frac{d^{3}p}{{\left(2\pi \right)}^3}}{\displaystyle \frac{1}{2{\omega }_p}}\left(e^{-ipx}{a}_p+e^{ipx}{a}_p^{†}\right),{\omega }_p=\sqrt{{\overrightarrow{p}}^2+m^2},px=p_0{x}_0-\overrightarrow{p}·\overrightarrow{x}\hfill \\ & & [a_p,a_q^{†}]={\left(2\pi \right)}^3\left(2{\omega }_p\right){\delta }^3(p-q),[a_p,a_q]=0.\hfill \end{array}$$

(9)

Let us proceed by providing the derivation of the unitary evolution operator of Equation (7). One starts with the Hamiltonian

$$H=H_0+H_I\left(t\right),$$

(10)

where $H_0$ stands for the free Hamiltonian

$$H_0=\sigma (J_z^A\otimes {⊮}^B+J_z^B\otimes {⊮}^A)+\int d\mu \left(p\right){\omega }_p{a}_p^{†}{a}_p,d\mu \left(p\right)={\displaystyle \frac{d^{3}p}{{\left(2\pi \right)}^3}}{\displaystyle \frac{1}{2{\omega }_p}},$$

(11)

and $H_I\left(t\right)$ is the interaction term:

$$H_I\left(t\right)=J_z^A\int d^{3}x\phi (x,0)f_A(x,t)+J_z^B\int d^{3}x\phi (x,0)f_B(x,t).$$

(12)

Notice that

$$H_0{|\psi \rangle }_{AB}\otimes |0\rangle =0.$$

(13)

For the evolution operator in the interaction representation, we have

$$\mathcal{U}\left(t\right)=T_t{e}^{-i{\int }^{t}d\tau H_I\left(\tau \right)},$$

(14)

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

$$T_t{e}^{{\int }^{t}d\tau A\left(\tau \right)}=e^{\Omega \left(t\right)},\Omega \left(t\right)=\sum _{n=1}^{\infty }{\Omega }_n\left(t\right)$$

(15)

with

$$\begin{array}{ccc}\hfill {\Omega }_1\left(t\right)& =& {\int }^{t}d\tau A\left(\tau \right)\hfill \\ \hfill {\Omega }_2\left(t\right)& =& {\displaystyle \frac{1}{2}}{\int }^{t}d{\tau }_1{\int }^{{\tau }_1}d{\tau }_2[A\left({\tau }_1\right),A\left({\tau }_2\right)]\hfill \\ \hfill {\Omega }_j& =& \mathrm{higher}\mathrm{order}\mathrm{commutators}j\ge 3.\hfill \end{array}$$

(16)

We must remember that the field commutator $\left[\phi \right(x),\phi (y\left)\right]$ is a c-number. As a consequence, ${\Omega }_j=0,i\ge 3$, while ${\Omega }_2$ yields an irrelevant phase. Therefore, up to an irrelevant phase, for the evolution operator $\mathcal{U}\left(t\right)$, one has

$$\mathcal{U}\left(t\right)=e^{-i{\int }^{t}d\tau H_I\left(\tau \right)}.$$

(17)

Therefore, in the large asymptotic time, and $t\to \infty$, Equation (7) follows; namely, after a long period of time, the density matrix can be written as

$$\begin{array}{c}\hfill {\rho }_{AB\phi }={\rho }_{AB\phi }(t\to \infty )=\mathcal{U}{\rho }_{AB\phi }\left(0\right){\mathcal{U}}^{†}.\end{array}$$

(18)

The subsequent stage involves deriving the density matrix ${\widehat{\rho }}_{AB}$ for the qutrit system through the process of tracing out the field modes:

$${\widehat{\rho }}_{AB}={\mathrm{Tr}}_{\phi }\left({\rho }_{AB\phi }\right).$$

(19)

Finally, once the density matrix ${\widehat{\rho }}_{AB}$ is known, one is capable of evaluating the Bell–CHSH correlator

$$\langle \mathcal{C}\rangle =\mathrm{Tr}\left({\widehat{\rho }}_{AB}\mathcal{C}\right),$$

(20)

where

$$\mathcal{C}=(A+A^{\prime })\otimes B+(A-A^{\prime })\otimes B^{\prime },$$

(21)

with $(A,A^{\prime })$, $(B,B^{\prime })$ being the Bell operators, namely

$$\begin{array}{ccc}\hfill A& =& A^{†},A^{\prime }=A^{\prime †},B=B^{†},B^{\prime }=B^{\prime †}\hfill \\ \hfill A^2& =& A^{\prime 2}=B^2=B^{\prime 2}=1\hfill \\ \hfill \left[A,B\right]& =& [A,B^{\prime }]=[A^{\prime },B]=[A^{\prime },B^{\prime }]=0.\hfill \end{array}$$

(22)

Concerning the commutators $[A,A^{\prime }]$ and $[B,B^{\prime }]$, they can be expressed in terms of the four Bell’s parameters $(\alpha ,{\alpha }^{\prime },\beta ,{\beta }^{\prime })$ in Equation (63), i.e.,

$$[A,A^{\prime }]=\left(\begin{array}{ccc}2sin(\alpha -{\alpha }^{\prime })& 0& 0\\ 0& 0& 0\\ 0& 0& 2sin({\alpha }^{\prime }-\alpha )\end{array}\right)$$

(23)

and

$$[B,B^{\prime }]=\left(\begin{array}{ccc}2sin({\beta }^{\prime }-\beta )& 0& 0\\ 0& 0& 0\\ 0& 0& 2sin(\beta -{\beta }^{\prime })\end{array}\right)$$

(24)

The Bell–CHSH inequality is said to be violated whenever

$$2<\left|\langle \mathcal{C}\rangle \right|\le 2\sqrt{2}.$$

(25)

2. Evaluation of the Qutrit Density Matrix in the Case of the Dephasing Coupling Detectors

We shall consider the density matrix ${\widehat{\rho }}_{AB}$ in the so-called dephasing coupling regime [2,3], for which the evolution operator is given by $\mathcal{U}={\mathcal{U}}_A\otimes {\mathcal{U}}_B$, where the unitary operator for the detector $j=A,B$ is

$$\begin{array}{c}\hfill {\mathcal{U}}_j=e^{-i{J}_j^z\otimes \phi \left(f_j\right)},\end{array}$$

(26)

with the commutation relation

$$\begin{array}{c}\hfill \left[{\mathcal{U}}_A,{\mathcal{U}}_B\right]=0.\end{array}$$

(27)

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 several practical simplifications to be carried out in the evaluation of the resulting density matrix for the qutrits system.

Using the algebra of the spin 1 matrices, expression (26) can be written as

$$\begin{array}{c}\hfill {\mathcal{U}}_j={⊮}_j-i{J}_j^z{s}_j+{\left(J_j^z\right)}^2(c_j-1),\end{array}$$

(28)

where $c_j\equiv cos\phi \left(f_j\right)$ and $s_j\equiv sin\phi \left(f_j\right)$. The evolution of the initial density matrix ${\rho }_{AB\phi }\left(0\right)$ given in Equation (5) is described as follows:

$$\begin{array}{cc}\hfill {\rho }_{AB\phi }=& \left({\mathcal{U}}_A\otimes {\mathcal{U}}_B\right){\rho }_{AB\phi }\left(0\right)({\mathcal{U}}_A^{†}\otimes {\mathcal{U}}_B^{†})\hfill \\ \hfill =& \left[{⊮}_A-i{J}_A^z{s}_A{\left(J_A^z\right)}^2(c_A-1)\right]\otimes \left[{⊮}_B-i{J}_B^z{s}_B{\left(J_B^z\right)}^2(c_B-1)\right]{\rho }_{AB}\left(0\right)|0\rangle \langle 0|\hfill \\ \hfill & \times \left[{⊮}_A+i{J}_A^z{s}_A{\left(J_A^z\right)}^2(c_A-1)\right]\otimes \left[{⊮}_B+i{J}_B^z{s}_B{\left(J_B^z\right)}^2(c_B-1)\right].\hfill \end{array}$$

(29)

Tracing over $\phi$, we obtain a rather lengthy expression for ${\widehat{\rho }}_{AB}$, namely

$$\begin{array}{cc}\hfill {\widehat{\rho }}_{AB}=& {\rho }_{AB}\left(0\right)+{\rho }_{AB}\left(0\right){\left(J_B^z\right)}^2\langle c_B-1\rangle -{\rho }_{AB}\left(0\right)J_A^z\otimes J_B^z\langle s_A{s}_B\rangle +{\rho }_{AB}\left(0\right){\left(J_A^z\right)}^2\langle c_A-1\rangle \hfill \\ \hfill & +{\rho }_{AB}\left(0\right){\left(J_A^z\right)}^2\otimes {\left(J_B^z\right)}^2\langle (c_A-1)(c_B-1)\rangle +{\left(J_B^z\right)}^2{\rho }_{AB}\left(0\right)J_B^z\langle s_B^2\rangle +{\left(J_B^z\right)}^2{\rho }_{AB}\left(0\right)J_A^z\langle s_B{s}_A\rangle \hfill \\ \hfill & +J_B^z{\rho }_{AB}\left(0\right)J_A^z\otimes {\left(J_B^z\right)}^2\langle s_B{s}_A(c_B-1)\rangle +J_B^z{\rho }_{AB}\left(0\right){\left(J_A^z\right)}^2\otimes J_B^z\langle s_B^2(c_A-1)\rangle \hfill \\ \hfill & +{\left(J_B^z\right)}^2{\rho }_{AB}\left(0\right)\langle c_B-1\rangle +{\left(J_B^z\right)}^2{\rho }_{AB}\left(0\right){\left(J_B^z\right)}^2\langle {(c_B-1)}^2\rangle -{\left(J_B^z\right)}^2{\rho }_{AB}\left(0\right)J_z^B\otimes J_z^A\langle (c_B-1)s_A{s}_B\rangle \hfill \\ \hfill & +{\left(J_B^z\right)}^2{\rho }_{AB}\left(0\right)J_z^A\langle (c_B-1)(c_A-1)\rangle +{\left(J_B^z\right)}^2{\rho }_{AB}\left(0\right){\left(J_B^z\right)}^2\otimes {\left(J_A^z\right)}^2\langle {(c_B-1)}^2(c_A-1)\rangle \hfill \\ \hfill & +J_A^z{\rho }_{AB}\left(0\right)J_B^z\langle s_A{s}_B\rangle +J_A^z{\rho }_{AB}\left(0\right)J_A^z\langle s_A^2\rangle +J_A^z{\rho }_{AB}\left(0\right)J_A^z\otimes {\left(J_B^z\right)}^2\langle s_A^2(c_B-1)\rangle \hfill \\ \hfill & +J_A^z{\rho }_{AB}\left(0\right){\left(J_A^z\right)}^2\otimes J_B^z\langle s_A{s}_B(c_A-1)\rangle -J_A^z\otimes J_B^z{\rho }_{AB}\left(0\right)\langle s_A{s}_B\rangle -J_A^z\otimes J_B^z{\rho }_{AB}\left(0\right){\left(J_B^z\right)}^2\langle s_A{s}_B(c_B-1)\rangle \hfill \\ \hfill & +J_A^z\otimes J_B^z{\rho }_{AB}\left(0\right)J_A^z\otimes J_B^z\langle s_A^2{s}_B^2\rangle -J_A^z\otimes J_B^z{\rho }_{AB}\left(0\right){\left(J_A^z\right)}^2\langle s_A{s}_B(c_A-1)\rangle \hfill \\ \hfill & -J_A^z\otimes J_B^z{\rho }_{AB}\left(0\right){\left(J_A^z\right)}^2\otimes {\left(J_B^z\right)}^2\langle s_A{s}_B(c_A-1)(c_B-1)\rangle +J_A^z\otimes {\left(J_B^z\right)}^2{\rho }_{AB}\left(0\right)J_B^z\langle s_A{s}_B(c_B-1)\rangle \hfill \\ \hfill & +J_A^z\otimes {\left(J_B^z\right)}^2{\rho }_{AB}\left(0\right)J_A^z\langle s_A^2(c_B-1)\rangle +J_A^z\otimes {\left(J_B^z\right)}^2{\rho }_{AB}\left(0\right)J_A^z\otimes {\left(J_B^z\right)}^2\langle s_A^2{(c_B-1)}^2\rangle \hfill \\ \hfill & +J_A^z\otimes {\left(J_B^z\right)}^2{\rho }_{AB}\left(0\right){\left(J_A^z\right)}^2\otimes J_B^z\langle s_A{s}_B(c_B-1)(c_A-1)\rangle +{\left(J_A^z\right)}^2{\rho }_{AB}\left(0\right)\langle (c_A-1)\rangle \hfill \\ \hfill & +{\left(J_A^z\right)}^2{\rho }_{AB}\left(0\right){\left(J_B^z\right)}^2\langle (c_A-1)(c_B-1)\rangle -{\left(J_A^z\right)}^2{\rho }_{AB}\left(0\right)J_A^z\otimes J_B^z\langle (c_A-1)s_A{s}_B\rangle \hfill \\ \hfill & +{\left(J_A^z\right)}^2{\rho }_{AB}\left(0\right){\left(J_A^z\right)}^2\langle {(c_A-1)}^2\rangle +{\left(J_A^z\right)}^2{\rho }_{AB}\left(0\right){\left(J_A^z\right)}^2\otimes {\left(J_B^z\right)}^2\langle {(c_A-1)}^2(c_B-1)\rangle \hfill \\ \hfill & +{\left(J_A^z\right)}^2\otimes J_B^z{\rho }_{AB}\left(0\right)J_B^z\langle (c_A-1)s_B^2\rangle +{\left(J_A^z\right)}^2\otimes J_B^z{\rho }_{AB}\left(0\right)J_A^z\langle (c_A-1)s_B{s}_A\rangle \hfill \\ \hfill & +{\left(J_A^z\right)}^2\otimes J_B^z{\rho }_{AB}\left(0\right)J_A^z\otimes {\left(J_B^z\right)}^2\langle (c_A-1)(c_B-1)s_B{s}_A\rangle +{\left(J_A^z\right)}^2\otimes J_B^z{\rho }_{AB}\left(0\right){\left(J_A^z\right)}^2\otimes J_B^z\langle {(c_A-1)}^2{s}_B^2\rangle \hfill \\ \hfill & +{\left(J_A^z\right)}^2\otimes {\left(J_B^z\right)}^2{\rho }_{AB}\left(0\right)\langle (c_A-1)(c_B-1)\rangle +{\left(J_A^z\right)}^2\otimes J_B^z{\rho }_{AB}\left(0\right){\left(J_B^z\right)}^2\langle (c_A-1){(c_B-1)}^2\rangle \hfill \\ \hfill & -{\left(J_A^z\right)}^2\otimes J_B^z{\rho }_{AB}\left(0\right)J_A^z\otimes J_B^z\langle (c_A-1)(c_B-1)s_A{s}_B\rangle +{\left(J_A^z\right)}^2\otimes J_B^z{\rho }_{AB}\left(0\right){\left(J_A^z\right)}^2\langle {(c_A-1)}^2(c_B-1)\rangle \hfill \\ \hfill & +{\left(J_A^z\right)}^2\otimes J_B^z{\rho }_{AB}\left(0\right){\left(J_A^z\right)}^2\otimes {\left(J_B^z\right)}^2\langle {(c_A-1)}^2{(c_B-1)}^2\rangle .\hfill \end{array}$$

(30)

where $\langle s_A{s}_B(c_A-1)(c_B-1)\rangle$, etc., denotes the expectation value of

$$\langle s_A{s}_B(c_A-1)(c_B-1)\rangle =\langle 0|s_A{s}_B(c_A-1)(c_B-1)|0\rangle ,$$

(31)

where

$$s_A={\displaystyle \frac{1}{2i}}(e^{i\phi \left(f_A\right)}-e^{-i\phi \left(f_A\right)}).$$

(32)

In the next section we shall see how these correlation functions can be addressed in the closed form, using the Tomita–Takesaki theory.

3. Tomita–Takesaki Modular Theory and the von Neumann Algebra of the Weyl Operators

To calculate the correlation functions of the Weyl operators shown in Equation (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. [19,21].

Let us begin by recalling the commutator between the scalar fields for arbitrary spacetime separation

$$\begin{array}{c}\hfill [\phi \left(x\right),\phi \left(y\right)]=i{\Delta }_{PJ}(x-y),\end{array}$$

(33)

where the Lorentz-invariant causal Pauli–Jordan distribution ${\Delta }_{PJ}(x-y)$ is defined by

$$\begin{array}{c}\hfill i{\Delta }_{PJ}(x-y)=\int {\displaystyle \frac{d^{4}k}{{\left(2\pi \right)}^3}}\epsilon \left(k^0\right)\delta (k^2-m^2)e^{-ik(x-y)},\end{array}$$

(34)

where $\epsilon \left(x\right)\equiv \theta \left(x\right)-\theta (-x)$. The Pauli–Jordan distribution ${\Delta }_{PJ}(x-y)$ vanishes outside of the light cone, guaranteeing that measurements at points separated by space-like intervals do not interfere; that is

$${\Delta }_{PJ}(x-y)=0,\mathrm{for}{(x-y)}^2<0.$$

(35)

Now, let $\mathcal{O}$ be a subregion of the ${\mathcal{W}}^R$ and let $\mathcal{M}\left(\mathcal{O}\right)$ be the space of smooth test functions with support contained in $\mathcal{O}$, namely

$$\begin{array}{c}\hfill \mathcal{M}\left(\mathcal{O}\right)=\left\{f\right|supp\left(f\right)\subseteq \mathcal{O}\}.\end{array}$$

(36)

Following [11,12], one introduces the symplectic complement of $\mathcal{M}\left(\mathcal{O}\right)$ as

$$\begin{array}{c}\hfill {\mathcal{M}}^{\prime }\left(\mathcal{O}\right)=\left\{g\right|{\Delta }_{PJ}(g,f)=0,\forall f\in \mathcal{M}\left(\mathcal{O}\right)\}.\end{array}$$

(37)

This symplectic complement ${\mathcal{M}}^{\prime }\left(\mathcal{O}\right)$ comprises all test functions for which the smeared Pauli–Jordan expression ${\Delta }_{PJ}(f,g)$ vanishes for any f belonging to $\mathcal{M}\left(\mathcal{O}\right)$,

$$\left[\phi \left(f\right),\phi \left(g\right)\right]=i{\Delta }_{PJ}(f,g),$$

(38)

allowing us to rephrase causality, Equation (35), as [11,12]

$$\begin{array}{c}\hfill \left[\phi \left(f\right),\phi \left(g\right)\right]=0,\end{array}$$

(39)

whenever $f\in \mathcal{M}\left(\mathcal{O}\right)$ and $g\in {\mathcal{M}}^{\prime }\left(\mathcal{O}\right)$.

As already mentioned in Section 1, the so-called Weyl operators [11,12,19] play an important role in the study of the Bell–CHSH inequality. This class of unitary operators is obtained by exponentiating the smeared field

$$W_h=e^{i\phi \left(h\right)}.$$

(40)

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:

$$\begin{array}{cc}\hfill W_f{W}_g& =e^{-{\displaystyle \frac{i}{2}}{\Delta }_{\mathrm{PJ}}(f,g)}{W}_{(f+g)},\hfill \\ \hfill W_f^{†}{W}_f& =W_f{W}_f^{†}=1,\hfill \\ \hfill W_f^{†}& =W_{(-f)}.\hfill \end{array}$$

(41)

Moreover, if f and g are space-like, the Weyl operators $W_f$ and $W_g$ commute. By expanding the field $\phi$ in terms of creation and annihilation operators, one can evaluate the expectation value of the Weyl operator, finding

$$\langle 0|W_h|0\rangle =e^{-{\displaystyle \frac{1}{2}}{\parallel h\parallel }^2},$$

(42)

where ${\left|\right|h\left|\right|}^2=\langle h|h\rangle$ and

$$\begin{array}{ccc}\hfill \langle f|g\rangle & =& \int {\displaystyle \frac{d^{3}k}{{\left(2\pi \right)}^3}}{\displaystyle \frac{1}{2{\omega }_k}}f{({\omega }_k,\overrightarrow{k})}^{*}g({\omega }_k,\overrightarrow{k}),\hfill \end{array}$$

(43)

is the Lorentz invariant inner product between the test functions $(f,g)$ [11,12,19] with the usual relation ${\omega }_k^2={\overrightarrow{k}}^2+m^2$ and

$$\begin{array}{c}\hfill f({\omega }_k,\overrightarrow{k})=\int d^{4}x{e}^{ikx}f\left(x\right),k_0={\omega }_k.\end{array}$$

A von Neumann algebra $\mathcal{A}\left(\mathcal{M}\right)$ arises by taking all possible products and linear combinations of the Weyl operators defined on $\mathcal{M}\left(\mathcal{O}\right)$. In particular, the Reeh–Schlieder theorem [11,12,17,20] states that the vacuum state $|0\rangle$ is both cyclic and separate for the von Neumann algebra $\mathcal{A}$. Consequently, we can apply the Tomita–Takesaki modular theory [11,12,17,18,19] and introduce the anti-linear unbounded operator S, whose action on the von Neumann algebra $\mathcal{A}\left(\mathcal{M}\right)$ is defined as

$$\begin{array}{c}\hfill Sa|0\rangle =a^{†}|0\rangle ,\forall a\in \mathcal{A}\left(\mathcal{M}\right),\end{array}$$

(44)

from which it follows that $S^2=1$ and $S|0\rangle =|0\rangle$. The operator S has a unique polar decomposition [18]:

$$\begin{array}{c}\hfill S=J{\Delta }^{1/2},\end{array}$$

(45)

where J is anti-unitary and $\Delta$ is positive and self-adjoint. These operators are characterized by the following set of properties [11,12,17,18,19]:

$$\begin{array}{ccc}\hfill \Delta & =& S^{†}S,J{\Delta }^{1/2}J={\Delta }^{-1/2},\hfill \\ \hfill J^2& =& 1,S^{†}=J{\Delta }^{-1/2},\hfill \\ \hfill J^{†}& =& J,{\Delta }^{-1}=S{S}^{†}.\hfill \end{array}$$

(46)

From the Tomita–Takesaki theorem [11,12,17,18,19], it follows that $J\mathcal{A}\left(\mathcal{M}\right)J={\mathcal{A}}^{\prime }\left(\mathcal{M}\right)$, meaning that, upon conjugation by the operator J, the algebra $\mathcal{A}\left(\mathcal{M}\right)$ is mapped onto its commutant ${\mathcal{A}}^{\prime }\left(\mathcal{M}\right)$, namely:

$${\mathcal{A}}^{\prime }\left(\mathcal{M}\right)=\left\{a^{\prime }\right|[a,a^{\prime }]=0,\forall a\in \mathcal{A}\left(\mathcal{M}\right)\}.$$

(47)

Furthermore, the theorem states that there is a one-parameter family of operators ${\Delta }^{it}$, $t\in \mathbb{R}$, that leave the algebra $\mathcal{A}\left(\mathcal{M}\right)$ invariant, such that the the following equation holds

$$\begin{array}{c}\hfill {\Delta }^{it}\mathcal{A}\left(\mathcal{M}\right){\Delta }^{-it}=\mathcal{A}\left(\mathcal{M}\right).\end{array}$$

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 [19], there is 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 $B_f=J{A}_{f}J$, ensuring their mutual commutativity due to the Tomita–Takesaki theorem, as $B_f=J{A}_{f}J$ belongs to the commutant ${\mathcal{A}}^{\prime }\left(\mathcal{M}\right)$ [19].

An important outcome of the Tomita–Takesaki modular theory, established by [22,23], allows the extension of the action of the modular operators $(J,\Delta )$ to the space of the test functions. In fact, when equipped with the Lorentz-invariant inner product $\langle f|g\rangle$, Equation (43), the set of test functions forms a complex Hilbert space $\mathcal{F}$ that possesses a variety of properties. To be more precise, it is found that the subspaces $\mathcal{M}$ and $i\mathcal{M}$ are standard subspaces for $\mathcal{F}$ [22]. This implies that:

i.

$\mathcal{M}\cap i\mathcal{M}=\left\{0\right\}$;

ii.

$\mathcal{M}+i\mathcal{M}$ is dense in $\mathcal{F}$.

As shown in [22], for such subspaces, it is viable to establish a modular theory similar to that of the Tomita–Takesaki theory. This involves introducing an operator s acting on $\mathcal{M}+i\mathcal{M}$ such that

$$\begin{array}{c}\hfill s(f+ih)=f-ih,\end{array}$$

(48)

for $f,h\in \mathcal{M}$. With this definition, it is worth noting that $s^2=1$. Employing the polar decomposition, one obtains:

$$\begin{array}{c}\hfill s=j{\delta }^{1/2},\end{array}$$

(49)

where j is an anti-unitary operator, while $\delta$ is positive and self-adjoint. Similarly to the operators $(J,\Delta )$, the operators $(j,\delta )$ fulfill the following properties [22]:

$$\begin{array}{cc}\hfill j{\delta }^{1/2}j& ={\delta }^{-1/2},{\delta }^{†}=\delta ,\hfill \\ \hfill s^{†}& =j{\delta }^{-1/2},j^{†}=j;\hfill \\ \hfill \delta & =s^{†}s,j^2=1.\hfill \end{array}$$

(50)

Further, one can show [12,22] that a test function f belongs to $\mathcal{M}$ if and only if

$$sf=f.$$

(51)

Indeed, let us suppose that $f\in \mathcal{M}$. From Equation (48), one can express

$$sf=h_1+i{h}_2,$$

(52)

for some $(h_1,h_2)$. Since $s^2=1$ it follows that

$$f=s(h_1+i{h}_2)=h_1-i{h}_2,$$

(53)

so that $h_1=f$ and $h_2=0$. Similarly, one finds that $f^{\prime }\in {\mathcal{M}}^{\prime }$ if and only if $s^{†}{f}^{\prime }=f^{\prime }$.

Thus, the lifting of the action of the operators $(J,\Delta )$ to the space of test functions is accomplished by [23]

$$\begin{array}{c}\hfill J{e}^{i\phi \left(f\right)}J=e^{-i\phi \left(jf\right)},\Delta e^{i\phi \left(f\right)}{\Delta }^{-1}=e^{i\phi \left(\delta f\right)}.\end{array}$$

(54)

Also, it is important to note that if $f\in \mathcal{M}\Rightarrow jf\in {\mathcal{M}}^{\prime }$. This property follows from

$$s^{†}\left(jf\right)=j{\delta }^{-1/2}jf=\delta f=j\left(j\delta f\right)=j\left(sf\right)=jf.$$

(55)

It is worth reminding here that $\delta$ is an unbounded operator with a continuous spectrum. For instance, as one learns from the work of [24], for the wedge ${\mathcal{W}}^R$, the spectrum of $\delta$ coincides with the positive real line, i.e., $log\left(\delta \right)=\mathbb{R}$. In the case of a continuous spectrum we lack the notion of eigenstates. Rather, it is appropriate to make use of the spectral decomposition [18] of the operator $\delta$ and refer to spectral subspaces ${\sigma }_{\lambda }$, parametrized by a real parameter $\lambda \in {\mathbb{R}}_{+}$.

We now have 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

$$\langle e^{i\phi \left(f_A\right)}{e}^{\pm i\phi \left(f_B\right)}\rangle =\langle e^{i(\varphi \left(f_A\right)\pm \varphi \left(f_B\right))}\rangle =e^{-\frac{1}{2}\left|\right|f_A\pm f_B{\left|\right|}^2},$$

(56)

so that we need to evaluate the following norms $\left(\right|\left|f_A\right|{|}^2,\left|\right|f_B\left|{|}^2\right)$ and the inner products $\langle f_A|f_B\rangle$. We begin by focusing on Alice’s test function $f_A$. We require that $f_A\in \mathcal{M}\left(\mathcal{O}\right)$ where $\mathcal{O}$ is located in the right Rindler wedge. Following [11,12,19], the test function $f_A$ can be further specified by considering the spectrum of the operator $\delta$. By selecting the subspace ${\sigma }_{\lambda }=[{\lambda }^2-\epsilon ,{\lambda }^2+\epsilon ]$ and introducing the normalized vector $\varphi$ belonging to this subspace, one writes

$$f_A=\eta (1+s)\varphi ,$$

(57)

where $\eta$ is an arbitrary parameter. As required by the setup outlined above, Equation (57) ensures that

$$s{f}_A=f_A.$$

(58)

We observe that $j\varphi$ is orthogonal to $\varphi$, i.e., $\langle \varphi |j\varphi \rangle =0$. In fact, from

$$\begin{array}{c}\hfill {\delta }^{-1}\left(j\varphi \right)=j\left(j{\delta }^{-1}j\right)\varphi =j\left(\delta \varphi \right),\end{array}$$

(59)

it follows that the modular conjugation j exchanges the spectral subspace $[{\lambda }^2-\epsilon ,{\lambda }^2+\epsilon ]$ with $[1/{\lambda }^2-\epsilon ,1/{\lambda }^2+\epsilon ]$. Regarding Bob’s test function $f_B$, we use the modular conjugation operator j and define

$$f_B=j{f}_A,$$

(60)

ensuring that

$$s^{†}{f}_B=f_B$$

(61)

This implies that, as required by the relativistic causality, $f_B$ belongs to the symplectic complement ${\mathcal{M}}^{\prime }\left(\mathcal{O}\right)$, located in the left Rindler wedge, namely: $f_B\in {\mathcal{M}}^{\prime }\left(\mathcal{O}\right)$. Finally, considering that $\varphi$ belongs to the spectral subspace $[{\lambda }^2-\epsilon ,{\lambda }^2+\epsilon ]$, it follows that [19],

$$\begin{array}{cc}\hfill \left|\right|f_A{\left|\right|}^2& =\left|\right|j{f}_A{\left|\right|}^2={\eta }^2(1+{\lambda }^2),\hfill \\ \hfill \langle f_A|j{f}_A\rangle & =2{\eta }^2\lambda ,\hfill \end{array}$$

(62)

which provides us with the necessary inner products.

4. The Bell–CHSH Inequality

We now face the Bell–CHSH inequality shown in Equation (20). We begin by defining the Bell operators [11,12]:

$$\begin{array}{cccccc}\hfill A|-1\rangle & =e^{i\alpha }|1\rangle ,\hfill & \hfill A|0\rangle & =|0\rangle ,\hfill & \hfill A|1\rangle & =e^{-i\alpha }|-1\rangle \hfill \\ \hfill A^{\prime }|-1\rangle & =e^{i{\alpha }^{\prime }}|1\rangle ,\hfill & \hfill A^{\prime }|0\rangle & =|0\rangle ,\hfill & \hfill A^{\prime }|1\rangle & =e^{-i{\alpha }^{\prime }}|-1\rangle \hfill \\ \hfill B|-1\rangle & =e^{-i\beta }|1\rangle ,\hfill & \hfill B|0\rangle & =|0\rangle ,\hfill & \hfill B|1\rangle & =e^{i\beta }|-1\rangle \hfill \\ \hfill B^{\prime }|-1\rangle & =e^{-i{\beta }^{\prime }}|1\rangle ,\hfill & \hfill B^{\prime }|0\rangle & =|0\rangle ,\hfill & \hfill B^{\prime }|1\rangle & =e^{i{\beta }^{\prime }}|-1\rangle ,\hfill \end{array}$$

(63)

which fulfill the whole set of conditions (22). The free parameters $(\alpha ,{\alpha }^{\prime },\beta ,{\beta }^{\prime })$, which will be chosen to be the most convenient, correspond to the four Bell’s angles.

We are reminded that the initial state for $AB$ is

$$\begin{array}{c}\hfill {|\psi \rangle }_{AB}={\displaystyle \frac{1}{\sqrt{3}}}\left({|-1\rangle }_A{\otimes |-1\rangle }_B-{|0\rangle }_A\otimes {|0\rangle }_B+{|1\rangle }_A\otimes {|1\rangle }_B\right),\end{array}$$

and using Equation (20), one obtains the Bell–CHSH correlator

$$\begin{array}{cc}\hfill \langle \mathcal{C}\rangle & ={\displaystyle \frac{1}{3}}[1+2cos(\alpha +\beta )]+{\displaystyle \frac{2}{3}}cos(\alpha +\beta )[2\langle (c_A-1)\rangle +2\langle (c_B-1)\rangle +4\langle s_A{s}_B\rangle \hfill \\ \hfill & -\langle s_A^2\rangle -\langle s_B^2\rangle +4\langle s_A{s}_B(c_B-1)\rangle +4\langle s_A{s}_B(c_A-1)\rangle -2\langle s_B^2(c_A-1)\rangle \hfill \\ \hfill & -2\langle s_A^2(c_B-1)\rangle +4\langle (c_A-1)(c_B-1)\rangle +\langle {(c_A-1)}^2\rangle +\langle {(c_B-1)}^2\rangle \hfill \\ \hfill & +2\langle (c_A-1){(c_B-1)}^2\rangle +2\langle {(c_A-1)}^2(c_B-1)\rangle +\langle s_A^2{s}_B^2\rangle \hfill \\ \hfill & +4\langle s_A{s}_B(c_A-1)(c_B-1)-\langle s_A^2{(c_B-1)}^2\rangle -\langle s_B^2{(c_A-1)}^2\rangle \hfill \\ \hfill & +\langle {(c_A-1)}^2{(c_B-1)}^2\rangle ]\hfill \\ \hfill & +{\displaystyle \frac{1}{3}}[1+2cos({\alpha }^{\prime }+\beta )]+{\displaystyle \frac{2}{3}}cos(\alpha +\beta )[2\langle (c_A-1)\rangle +2\langle (c_B-1)\rangle +4\langle s_A{s}_B\rangle \hfill \\ \hfill & -\langle s_A^2\rangle -\langle s_B^2\rangle +4\langle s_A{s}_B(c_B-1)\rangle +4\langle s_A{s}_B(c_A-1)\rangle -2\langle s_B^2(c_A-1)\rangle \hfill \\ \hfill & -2\langle s_A^2(c_B-1)\rangle +4\langle (c_A-1)(c_B-1)\rangle +\langle {(c_A-1)}^2\rangle +\langle {(c_B-1)}^2\rangle \hfill \\ \hfill & +2\langle (c_A-1){(c_B-1)}^2\rangle +2\langle {(c_A-1)}^2(c_B-1)\rangle +\langle s_A^2{s}_B^2\rangle \hfill \\ \hfill & +4\langle s_A{s}_B(c_A-1)(c_B-1)-\langle s_A^2{(c_B-1)}^2\rangle -\langle s_B^2{(c_A-1)}^2\rangle \hfill \\ \hfill & +\langle {(c_A-1)}^2{(c_B-1)}^2\rangle ]\hfill \\ \hfill & +{\displaystyle \frac{1}{3}}[1+2cos(\alpha +{\beta }^{\prime })]+{\displaystyle \frac{2}{3}}cos(\alpha +{\beta }^{\prime })[2\langle (c_A-1)\rangle +2\langle (c_B-1)\rangle +4\langle s_A{s}_B\rangle \hfill \\ \hfill & -\langle s_A^2\rangle -\langle s_B^2\rangle +4\langle s_A{s}_B(c_B-1)\rangle +4\langle s_A{s}_B(c_A-1)\rangle -2\langle s_B^2(c_A-1)\rangle \hfill \\ \hfill & -2\langle s_A^2(c_B-1)\rangle +4\langle (c_A-1)(c_B-1)\rangle +\langle {(c_A-1)}^2\rangle +\langle {(c_B-1)}^2\rangle \hfill \\ \hfill & +2\langle (c_A-1){(c_B-1)}^2\rangle +2\langle {(c_A-1)}^2(c_B-1)\rangle +\langle s_A^2{s}_B^2\rangle \hfill \\ \hfill & +4\langle s_A{s}_B(c_A-1)(c_B-1)-\langle s_A^2{(c_B-1)}^2\rangle -\langle s_B^2{(c_A-1)}^2\rangle \hfill \\ \hfill & +\langle {(c_A-1)}^2{(c_B-1)}^2\rangle ]\hfill \\ \hfill & -{\displaystyle \frac{1}{3}}[1+2cos({\alpha }^{\prime }+{\beta }^{\prime })]-{\displaystyle \frac{2}{3}}cos({\alpha }^{\prime }+{\beta }^{\prime })[2\langle (c_A-1)\rangle +2\langle (c_B-1)\rangle +4\langle s_A{s}_B\rangle \hfill \\ \hfill & -\langle s_A^2\rangle -\langle s_B^2\rangle +4\langle s_A{s}_B(c_B-1)\rangle +4\langle s_A{s}_B(c_A-1)\rangle -2\langle s_B^2(c_A-1)\rangle \hfill \\ \hfill & -2\langle s_A^2(c_B-1)\rangle +4\langle (c_A-1)(c_B-1)\rangle +\langle {(c_A-1)}^2\rangle +\langle {(c_B-1)}^2\rangle \hfill \\ \hfill & +2\langle (c_A-1){(c_B-1)}^2\rangle +2\langle {(c_A-1)}^2(c_B-1)\rangle +\langle s_A^2{s}_B^2\rangle \hfill \\ \hfill & +4\langle s_A{s}_B(c_A-1)(c_B-1)-\langle s_A^2{(c_B-1)}^2\rangle -\langle s_B^2{(c_A-1)}^2\rangle \hfill \\ \hfill & +\langle {(c_A-1)}^2{(c_B-1)}^2\rangle ].\hfill \end{array}$$

(64)

The expression above is written in terms of the inner products between test functions, which can be evaluated by employing the expressions (62). The final expression reads

$$\begin{array}{cc}\hfill \langle \mathcal{C}\rangle & ={\displaystyle \frac{2}{3}}\left\{1+[cos(\alpha +\beta )+cos({\alpha }^{\prime }+\beta )+cos(\alpha +{\beta }^{\prime })-cos({\alpha }^{\prime }+{\beta }^{\prime })]\right\}\hfill \\ \hfill & -{\displaystyle \frac{4}{3}}f(\eta ,\lambda )[cos(\alpha +\beta )+cos({\alpha }^{\prime }+\beta )+cos(\alpha +{\beta }^{\prime })-cos({\alpha }^{\prime }+{\beta }^{\prime })],\hfill \end{array}$$

(65)

where the function $f(\eta ,\lambda )$ is

$$\begin{array}{c}\hfill f(\eta ,\lambda )=e^{-2{\eta }^2(1+{\lambda }^2)}-e^{-4{\eta }^2{(1-\lambda )}^2},\end{array}$$

(66)

with $\eta \ne 0$. From (65), one learns several things:

• When the quantum field $\phi$ is removed, i.e., ${\eta }^2\to 0$, and therefore, $f(\eta ,\lambda )=0$, we recover the Bell–CHSH inequality of Quantum Mechanics for qutrits, whose maximum value is [14,15]

  $$\begin{array}{c}\hfill {\langle \mathcal{C}\rangle }_{f=0}={\displaystyle \frac{2}{3}}(1+2\sqrt{2})\approx 2.55>2.\end{array}$$

  (67)
  One notices that this value is lower than Tsirelson’s bound, as we are dealing with a spin 1 system.
• The contributions arising from the scalar field $\phi$ are encoded in the exponential terms $e^{-4{\eta }^2{(1-\lambda )}^2}$ and $e^{-2{\eta }^2(1+{\lambda }^2)}$. It is worth remembering here that the parameter ${\eta }^2$ is related to the norm of the test function $f_A$, Equation (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 [19,25], $\eta$ is a free parameter which appears 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 $(A,A^{\prime },B,B^{\prime })$, Equation (63), one realizes that they are dichotomic for arbitrary values of the parameters $(\alpha ,{\alpha }^{\prime },\beta ,{\beta }^{\prime })$. As such, they are completely free and, in fact, are chosen to be the most convenient in the final expression of the Bell–CHSH inequality. The same pattern is encountered in the case of the parameter $\eta$. One has to notice that the Weyl operator

  $$W_{f_A}=e^{i\phi \left(f_A\right)}=e^{i\phi \left(\eta \right(1+s\left)\varphi \right)},$$

  (68)

  is unitary for any value of the parameter $\eta$.
• We now have to face the choice of the spectral subspace ${\sigma }_{\lambda }$ of the modular operator $\delta$. This is a not an easy task due to the fact that $\delta$ has a continuous spectrum given by the positive real line ${\mathbb{R}}_{+}$. 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 field2, namely

  $${\displaystyle \frac{4\sqrt{2}\lambda }{1+{\lambda }^2}},$$

  (69)

  from which the choice of the spectral subspace $[\sqrt{2}-1,\sqrt{2}+1]$ follows. The maximum violation is attained for $\lambda =1$. The important point here is that this choice can be made, as $\lambda =1$ belongs to the continuous spectrum of $\delta$.
  In our case, we proceed as follows. From $e^{-2{\eta }^2(1+{\lambda }^2)}>e^{-4{\eta }^2(1-{\lambda }^2)}$, we obtain the roots $\lambda \pm =2\pm \sqrt{3}$. We can distinguish two possibilities. The first one is when $0<\lambda <2-\sqrt{3}=0.267$. In this case, the quantum field produces a damping, resulting in a decrease in the violation of the Bell–CHSH inequality, as compared to the pure Quantum Mechanical case. The second possibility takes place when $2-\sqrt{3}<\lambda <0.3$, resulting in an improvement of the size of the violation. In other words, we pick up the spectral subspace $[0,0.3]$. This spectral subspace has been identified through a numerical analysis. Again, it is a possible choice.
• Let us also check that the Tsirelson bound is fulfilled for arbitrary values of the parameter $\eta$. Observing that the maximum value of the angular part of Equation (65) is $2\sqrt{2}$, attained for the following values of the Bell’s angles:

  $$\begin{array}{ccc}\hfill \alpha & =& 0,{\alpha }^{\prime }={\displaystyle \frac{\pi }{2}},\beta =-{\displaystyle \frac{\pi }{4}},{\beta }^{\prime }={\displaystyle \frac{\pi }{4}},\hfill \end{array}$$

  (70)

  it follows that

  $$\begin{array}{c}\hfill \langle \mathcal{C}\rangle ={\displaystyle \frac{2}{3}}\{1+2\sqrt{2}[1-2f(\eta ,\lambda )]\}.\end{array}$$

  (71)
  Accordingly, the Tsirelson bound, $\langle \mathcal{C}\rangle \le 2\sqrt{2}$, is fulfilled whenever

  $$\begin{array}{c}\hfill f(\eta ,\lambda )\ge {\displaystyle \frac{1-\sqrt{2}}{4\sqrt{2}}},\end{array}$$

  (72)

  with $\lambda \in [0,0.3]$. A numerical investigation shows that Equation (72) is in fact fulfilled for arbitrary values of the normalization factor $0<\eta <\infty$3; see, e.g., Figure 1 for the behavior of $f(\eta ,\lambda )$ for $\eta \in [0,10]$.
• The whole effects produced by the quantum field can be seen in Figure 2. The orange surface represents the maximum value of $\langle \mathcal{C}\rangle$ without the presence of $\phi$, i.e., $\langle \mathcal{C}\rangle =2.55$. One notices the existence of a small region in blue, above the orange surface. This region corresponds to values of $(\eta ,\lambda )$ for which the size of the violation is improved, almost till approximately $2.7$. This phenomenon occurs when $0.3>\lambda >0.26$.

Comparison with the Spin $1/2$ Case

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

$$\left({\displaystyle \frac{{|+\rangle }_A{\otimes |+\rangle }_B{+|-\rangle }_A\otimes {|-\rangle }_B}{\sqrt{2}}}\right)\otimes |0\rangle .$$

(73)

The Bell–CHSH inequality is found

$${\langle \mathcal{C}\rangle }_{1/2}=2\sqrt{2}{e}^{-{\eta }^2{(1+\lambda )}^2},$$

(74)

which, unlike the case of the spin 1, exhibits only a decrease in the size of the violation.

5. Conclusions

In this work, we have analyzed the interaction between spin 1 Unruh–De Witt detectors, i.e., a pair of qutrits, and a relativistic quantum scalar field $\phi$. The effects of the quantum field on the Bell–CHSH inequality have been scrutinized in detail by making use of the dephasing channel for the evolution operator. By employing the Tomita–Takesaki modular theory and the properties of the Weyl operators, these effects have been evaluated in the closed form, as expressed by Equation (65).

The main finding of the present study is that the presence of a scalar quantum field may induce both a damping as well as an improving effect, resulting, respectively, in a decrease and an increase in the size of the violation of the Bell–CHSH inequality as compared to the case in which the field is absent.

As such, the case of spin 1 looks much different from that of spin $1/2$, for which only a decrease in the violation has been detected [10]. As has already been underlined, the existence of an improvement in the size of the violation of the Bell–CHSH inequality can be ascribed to the fact that, for spin 1, the Tsirelson bound $2\sqrt{2}$ is never saturated. Instead, the maximum value obtained in Quantum Mechanics is $[{\displaystyle \frac{2}{3}}(1+2\sqrt{2})\approx 2.55]$. As such, in the presence of a quantum field $\phi$, there exists a permissible interval, $[{\displaystyle \frac{2}{3}}(1+2\sqrt{2}),2\sqrt{2}]$, where an increase in the size of the violation occurs.