## Appendix A Phase-Mediated Entanglement in Quantum Field Theory
On the contrary, the strong \((a,b)\)-pair correlation is due to the coherent structure of the vacuum \(|0(\theta)\rangle\). In deriving Equation (A13), we have used the relation \((a_{\mu,\mathbf{k}}^{\dagger}a_{\mu,\mathbf{k}}-b_{-\mu,-\mathbf{k}}^{\dagger }b_{-\mu,-\mathbf{k}})|0(\theta)\rangle=0\) and \(n_{a_{\mu,\mathbf{k}}}(\theta)\equiv\langle\hat{n}_{a_{\mu,\mathbf{k}}}\rangle =\sinh^{2}\theta_{k}=\langle\hat{n}_{b_{-\mu,-\mathbf{k}}}\rangle\equiv n_{b_{ -\mu,-\mathbf{k}}}(\theta)\). Note that \((a_{\mu,\mathbf{k}}^{\dagger}a_{\mu,\mathbf{k}}-b_{-\mu,-\mathbf{k}}^{\dagger }b_{-\mu,-\mathbf{k}})\) is proportional to the Casimir operator of \(SU(1,1)\), and thus, it is a conserved quantity in all the state spaces. We finally remark that:  
\[H=H_{0}+W_{0}=\sum_{\mu}\int d^{3}kE_{k}(\alpha^{\dagger}_{\mu,\mathbf{k}} \alpha_{\mu,\mathbf{k}}+\beta^{\dagger}_{\mu,\mathbf{k}}\beta_{\mu,\mathbf{k} })+W_{0}. \tag{12}\]3 (A14)  
which shows that the measurement of the \(a\)-mode number performed by [PERSON] is in fact determined by [PERSON]'s \(\beta\)-modes: the strong phase correlation between the photon modes is dynamically built in in the \(|0(\theta)\rangle\) vacuum. In the experiments with, e.g., calcite crystals separating [PERSON]'s and [PERSON]'s respective photon beams into orthogonally-polarized beams, [PERSON]'s observation of the photon in one of the two polarization states determines [PERSON]'s observation of the (other) orthogonal polarization photon state, even if their observations occur at a space-like distance. These correlations are phase correlations, not-mediated by messenger particles, and therefore, there is no violation of relativity postulates. We also note that a single measurement made by [PERSON] determines only a probability distribution for the result observed by [PERSON]. Such a statistical feature is also built in in the condensate structure of \(|0(\theta)\rangle\). In fact, the \(a\)-mode (and \(b\)-mode) number is given by the Bose-Einstein distribution function:  
\[H=H_{0}+W_{0}=\sum_{\mu}\int d^{3}kE_{k}(\alpha^{\dagger}_{\mu,\mathbf{k}} \alpha_{\mu,\mathbf{k}}+\beta^{\dagger}_{\mu,\mathbf{k}}\beta_{\mu,\mathbf{k} })+W_{0}. \tag{12}\]4  
where \(\beta^{-1}=k_{B}T\), \(k_{B}\) is the Boltzmann constant and \(T\) the temperature. Equation (115) is obtained by the minimization condition of the free energy \(F_{a}\), \(\partial F_{a}/\partial\theta_{k}=0\), for all \(k\). The minimization of the free energy thus plays a crucial role in the entanglement phenomenon. The density matrix operator is given by \(\beta_{\mathbf{k},\mu}=f_{k}^{a_{\mu,k}^{\dagger}a_{\mu,k}}\), with \(f_{k}\equiv e^{-\beta\omega_{k}}\). Summing up, the observations by [PERSON] and [PERSON] are \"embedded\" in the collective mode vacuum background \(|0(\theta)\rangle\) controlled by the Bose-Einstein condensation. The \"element of reality\" in the entanglement problem is thus of the same degree as the one attributed to condensate states in condensed matter physics. [PERSON]'s measurements have a probabilistic effect on [PERSON]'s measurements at space-like separation, not induced by (spooky) forces, but due to the coherence of the vacuum condensate. Equation (115) shows that \(\theta_{k}=\theta_{k}(\beta)\), i.e., it is temperature dependent. We thus realize that \(|0(\theta)\rangle\) is actually a thermal state, and this leads us to consider the finite temperature QFT in the thermo field dynamics (TFD) formalism [5; 6], where the computation of thermal averages is performed in terms of the \(|0(\theta)\rangle\) pure state representation. In conclusion, phase-mediated long-range correlations over space-like distances find their origin in the coherent dynamical structure of the collective mode vacuum background. When entanglement is studied in the QFT formalism, in a natural way there are no \"spooky forces at a distance\" and violations of special relativity bounds.