## Appendix A Phase-Mediated Entanglement in Quantum Field Theory
Therefore, the locality of the QFT formalism is exactly the origin of the \"difference\" (unitary inequivalence, thus observable physical diversity) between the entangled \(|0(\theta)\rangle\) state and the non-entangled one \(|0\rangle\) (for the role of the infrared contributions in QFT, see, e.g., [93; 94; 95; 96]). These specific formal points represent distinctive features of QFT, and they mark the differences between the present discussion and preceding studies of the entanglement phenomenon. As already mentioned, these points and the results listed below are direct consequences of the existence in QFT of infinitely-many unitarily-inequivalent representations of the CCR. For brevity, we simply list the results obtained from Equation (A7) without reporting their derivation (details can be found in [5; 6; 7; 56; 91]). They are of general validity for such kinds of coherent states. The state (A7) is an entangled state for \(a_{\mu,\mathbf{k}}\) and \(b_{-\mu,-\mathbf{k}}\). In fact, it can be written as:  
\[|0(\theta)\rangle=\prod_{k}\frac{1}{\cosh\theta_{k}}\left[|0\rangle\otimes|0 \rangle+\sum_{\mathbf{k},\mu}\tanh\theta_{k}\left(|a_{\mu,\mathbf{k}}\rangle \otimes|b_{-\mu,-\mathbf{k}}\rangle\right)+\ldots\right],\] (A9)  
which shows that it cannot be factorized into the product of two single-mode states. In the summation term in Equation (A9), there are states of the Bell type, e.g., \(|\psi_{\pm}\rangle\propto\left(|a_{\mu,\mathbf{k}}\rangle\,|b_{-\mu,-\mathbf{ k}}\rangle\pm|a_{-\mu,\mathbf{k}}\rangle|b_{\mu,-\mathbf{k}}\rangle\right)\). Next, we introduce the entropy operator \(\mathcal{S}_{a}\), which provides a measure of the degree of entanglement:  
\[\mathcal{S}_{a}\equiv-\sum_{\mathbf{k},\mu}\Bigl{\{}a_{\mu,\mathbf{k}}^{\dagger }a_{\mu,\mathbf{k}}\ln\sinh^{2}\theta_{k}-a_{\mu,\mathbf{k}}a_{\mu,\mathbf{k}} ^{\dagger}\ln\cosh^{2}\theta_{k}\Bigr{\}}\.\] (A10)  
A similar expression for \(\mathcal{S}_{b}\) is obtained by replacing \(a_{\mu,\mathbf{k}}\) and \(a_{\mu,\mathbf{k}}^{\dagger}\) with \(b_{-\mu,-\mathbf{k}}\) and \(b_{-\mu,-\mathbf{k}}^{\dagger}\), respectively. We write \(\mathcal{S}\) for either \(\mathcal{S}_{a}\) or \(\mathcal{S}_{b}\). We obtain:  
\[H=H_{0}+W_{0}=\sum_{\mu}\int d^{3}kE_{k}(\alpha^{\dagger}_{\mu,\mathbf{k}} \alpha_{\mu,\mathbf{k}}+\beta^{\dagger}_{\mu,\mathbf{k}}\beta_{\mu,\mathbf{k} })+W_{0}. \tag{12}\]0 (A11)  
where \(W_{n}(\theta)\) is a decreasing monotonic function of \(n\), given by:  
\[H=H_{0}+W_{0}=\sum_{\mu}\int d^{3}kE_{k}(\alpha^{\dagger}_{\mu,\mathbf{k}} \alpha_{\mu,\mathbf{k}}+\beta^{\dagger}_{\mu,\mathbf{k}}\beta_{\mu,\mathbf{k} })+W_{0}. \tag{12}\]1 (A12)  
Equation (A11) shows that \(W_{n}\) gives the probability of having entanglement of the two sets of \(\{n\}\) modes \(a\) and \(b\) and confirms that \(\mathcal{S}\) provides a measure of the degree of entanglement. A further result is that the linear correlation coefficient \(J(\hat{n}_{a},\hat{n}_{b})\) acquires in \(|0(\theta)\rangle\) the maximal value for the \(\theta\)-phase-mediated correlation between the \(a\) and \(b\) modes, i.e.,  
\[H=H_{0}+W_{0}=\sum_{\mu}\int d^{3}kE_{k}(\alpha^{\dagger}_{\mu,\mathbf{k}} \alpha_{\mu,\mathbf{k}}+\beta^{\dagger}_{\mu,\mathbf{k}}\beta_{\mu,\mathbf{k} })+W_{0}. \tag{12}\]2 (A13)  
where for simplicity, the indexes \(\pm\mu\) and \(\pm\mathbf{k}\) are omitted, \(\hat{n}_{a}=a^{\dagger}a\), \(\hat{n}_{b}=b^{\dagger}b\) are the number operators, the variance is \(\langle(\Delta\hbar)^{2}\rangle\equiv\langle(\hbar-\langle\hat{n}\rangle)^{2} \rangle=\langle\hat{n}^{2}\rangle-\langle\hat{n}\rangle^{2}\), and the covariance is denoted by \(cov(\hat{n}_{a},\hat{n}_{b})\equiv\langle\hat{n}_{a}\hat{n}_{b}\rangle- \langle\hat{n}_{a}\rangle\)\(\langle\hat{n}_{b}\rangle\). The symbol \(\left\langle*\right\rangle\) denotes the expectation value in \(|0(\theta)\rangle\). For non-correlated modes \(\langle\hat{n}_{a}\hat{n}_{b}\rangle=\langle\hat{n}_{a}\rangle\)\(\langle\hat{n}_{b}\rangle\), the covariance is zero.