## Appendix A Phase-Mediated Entanglement in Quantum Field Theory
Thus, when it is written in terms of these physical fields, it must have the form:  
\[H=H_{0}+W_{0}=\sum_{\mu}\int d^{3}kE_{k}(\alpha^{\dagger}_{\mu,\mathbf{k}} \alpha_{\mu,\mathbf{k}}+\beta^{\dagger}_{\mu,\mathbf{k}}\beta_{\mu,\mathbf{k} })+W_{0}. \tag{12}\]One can show that Equation (A1) can be written in the form of Equation (A2) when \(E_{k}\) and \(W_{0}\) are given by:  
\[E_{k}=\sqrt{\omega_{k}^{2}-\
u_{k}^{2}}\,\qquad W_{0}=\int d^{3}k\,\left(\sqrt{ \omega_{k}^{2}-\
u_{k}^{2}}-\omega_{k}\right)\,\] (A3)  
and \(\alpha\) and \(\beta\) are expressed in terms of \(a\) and \(b\) as:  
\[\alpha_{\mu,\mathbf{k}}\equiv\alpha_{\mu,\mathbf{k}}(\theta)=a_{\mu,\mathbf{k }}\cosh\theta_{k}-b_{-\mu,-\mathbf{k}}^{\dagger}\sinh\theta_{k}\] (A4)  
\[\beta_{\mu,\mathbf{k}}\equiv\beta_{\mu,\mathbf{k}}(\theta)=b_{\mu,\mathbf{k }}\cosh\theta_{k}-a_{-\mu,-\mathbf{k}}^{\dagger}\sinh\theta_{k}\] (A5)  
and \(\theta_{k}\) given by:  
\[\cosh 2\theta_{k}=\frac{\omega_{k}}{\sqrt{\omega_{k}^{2}-\
u_{k}^{2}}}\,\qquad \sinh 2\theta_{k}=-\frac{\
u_{k}}{\sqrt{\omega_{k}^{2}-\
u_{k}^{2}}}\.\] (A6)  
Clearly, \(\alpha_{\mu,\mathbf{k}}\) and \(\beta_{\mu,\mathbf{k}}\) do not annihilate \(|0\rangle\). The state annihilated by \(\alpha_{\mu,\mathbf{k}}\) and \(\beta_{\mu,\mathbf{k}}\) is \(|0(\theta)\rangle\), \(\alpha_{\mu,\mathbf{k}}|0(\theta)\rangle=0,\ \beta_{\mu,\mathbf{k}}|0(\theta)\rangle=0\), given, at finite volume \(V\), by:  
\[|0(\theta)\rangle=\prod_{\mathbf{k},\mu}\frac{1}{\cosh\theta_{k}}\exp\left( \tanh\theta_{k}\,a_{\mu,\mathbf{k}}^{\dagger}b_{-\mu,-\mathbf{k}}^{\dagger} \right)|0\rangle,\] (A7)  
with \(\langle 0(\theta)|0(\theta)\rangle=1\). The state \(|0(\theta)\rangle\) is thus the physical vacuum state in the space \(\mathcal{H}_{\mathcal{F}}\), which now we denote as \(\mathcal{H}_{\mathcal{F}}(\theta)\). It is known [5; 6; 7; 23; 56; 91] to be a generalized \(SU(1,1)\) two-mode squeezed coherent state, with the phase \(\theta_{k}\) being related to the squeezing parameter. By using the continuous limit relation \(\sum_{\mathbf{k}}\to[V/(2\pi)^{3}]\int d^{3}k\) in Equation (A7), the factor \(\exp\left[(-V/2(2\pi)^{3})\int d^{3}k\,\mathcal{F}(\theta_{k}\right]\), with \(\mathcal{F}(\theta_{k})=\ln\cosh^{2}\theta_{k}\), appears, and for \(\int d^{3}k\,\mathcal{F}(\theta_{k})\) non-vanishing and finite, \(\mathcal{H}\) turns out to be unitarily inequivalent to \(\mathcal{H}_{\mathcal{F}}(\theta)\), for any \(\theta\equiv\{\theta_{k},\forall\,k\}\), for \(V\to\infty\), i.e.,  
\[\langle 0|0(\theta)\rangle\to 0\quad\text{for}\quad V\to\infty\,\] (A8)  
and similarly for any couple of states of \(\mathcal{H}\) and \(\mathcal{H}_{\mathcal{F}}(\theta)\); all the states of \(\mathcal{H}\) are orthogonal to those of \(\mathcal{H}_{\mathcal{F}}(\theta)\), unless \(\theta_{k}=0\) for all \(k\). The spaces \(\mathcal{H}_{\mathcal{F}}(\theta)\) and \(\mathcal{H}_{\mathcal{F}}(\theta^{\prime})\) are also unitarily inequivalent, \(\langle 0(\theta^{\prime})|0(\theta)\rangle\to 0\) for \(V\to\infty\), for any \(\theta\
eq\theta^{\prime}\). This is a characteristic feature of QFT, called foliation, not present in QM (the von Neumann theorem [92]). The unitary inequivalence expressed by Equation (A8) means that in the infinite volume limit, the state \(|0(\theta)\rangle\) cannot be expressed in terms of the state \(|0\rangle\). We have the transition to non-perturbative physics. In the \(V\to\infty\) limit, Equation (A7) is to be understood as a formal expression. Remarkably, the unitary inequivalence between \(|0(\theta)\rangle\) and \(|0\rangle\) arises since the contributions of the order of \((1/V)_{V\to\infty}\) are neglected. These infrared(or \((1/V)_{V\to\infty}\)) contributions are necessarily missing due to the \"locality\" of any laboratory observations, where the volume \(V\) can be always considered to be extremely large (infinite) with respect to the finiteness of the spatial region covered by the observations.