## 3 Entanglement in the Dissipative Quantum Model of Brain
We realize that the algebraic structure of the model is the one of TFD, and we also see that formal structures introduced in Appendix A are here re-obtained. The number \(\mathcal{N}_{A_{\kappa}}\) is given by:  
\[\mathcal{N}_{A_{\kappa}}=\mathcal{N}\langle 0|A_{\kappa}^{\dagger}A_{\kappa}|0 \rangle_{\mathcal{N}}=\sinh^{2}\theta_{\kappa}\, \tag{3}\]  
which shows how the \(\mathcal{N}\)-set, \(\mathcal{N}\equiv\{\mathcal{N}_{A_{\kappa}}=\mathcal{N}_{\tilde{A}_{\kappa}}, \forall\kappa,at\ \ t_{0}=0\}\), is related to the \(\theta\)-set, \(\theta\equiv\{\theta_{\kappa},\forall\kappa,at\ \ t_{0}=0\}\). For simplicity of notation, we use \(\mathcal{N}_{A_{\kappa}}\equiv\mathcal{N}_{A_{\kappa}}(\theta)\) and \(|0\rangle_{\mathcal{N}}\equiv|0(\theta)\rangle_{\mathcal{N}}\). We remark that for different codes \(\mathcal{N}\
eq\mathcal{N}^{\prime}\), \(\{|0\rangle_{\mathcal{N}}\}\) and \(\{|0\rangle_{\mathcal{N}^{\prime}}\}\) are each unitarily inequivalent state spaces in the infinite volume limit:  
\[{}_{\mathcal{N}}\langle 0|0\rangle_{\mathcal{N}^{\prime}}\underset{\mathcal{N} \rightarrow\infty}{\longrightarrow}0\quad\forall\,\mathcal{N}\
eq\mathcal{N}^ {\prime}\;. \tag{4}\]  
We conclude that representations \(\{|0\rangle_{\mathcal{N}}\}\) denote different, non-overlapping \"points\" with different labels \(\mathcal{N}\), meaning that \(\mathcal{N}\) is a good code. The space of the representations is the \"memory space\". We thus see that the possibility offered by QFT of having infinitely many unitarily-inequivalent representations \(\{|0\rangle_{\mathcal{N}}\}\), for all \(\mathcal{N}\)'s, of the CCR, provides a solution to the overprinting problem in the many-body model. Now, we have indeed that infinitely many vacua \(|0\rangle_{\mathcal{N}}\) for all \(\mathcal{N}\)'s are independently accessible, and memories of code \(\mathcal{N}\), for all \(\mathcal{N}\)'s, can be recorded, coexisting without destructive interference. We recognize that the state \(|0\rangle_{\mathcal{N}}\) is of the same type of the state in Equation (A7) in Appendix A, and in particular, it is an entangled state that cannot be factorized into two single-mode states (cf. Equation (A9)):  
\[|0\rangle_{\mathcal{N}}=\left(\prod_{\kappa}\ \frac{1}{\cosh\theta_{\kappa}} \right)\,\left(|0\rangle_{0}\otimes|\tilde{0}\rangle_{0}-\sum_{\kappa}\, \tanh\theta_{\kappa}\left(|A_{\kappa}\rangle\otimes|\tilde{A}_{\kappa}\rangle \right)+\ldots\right)\;. \tag{5}\]  
The conclusion at this point of the discussion is that the whole set of results listed in Equations (A8)-(A15) also holds in the present case. As stressed also in Appendix A, the features of the entanglement and the related results there presented are indeed of general validity, not limited to the special case of two photon entanglement, discussed there for the interested reader's convenience. Since the \(\tilde{A}\) modes represent the environment, we see that the brain-environment entanglement is a specific dynamical feature of the brain functional activity. The origin of the entanglement is dissipation. The entanglement robustness, rooted in the fact that in QFT there is no unitary operator able to disentangle the brain from its environment, ensures the stability and, at the same time, the persistence of the openness of brain on the world. All the above observations and remarks express the strict relationship between the formalism of the dissipative quantum model of brain and the entanglement phenomenon. As we see from the specific mathematical aspects discussed above, entanglement plays a central, essential role in the formal structure on which the dissipative model is constructed. We observe that, however, the robustness of the entanglement may be spoiled by finite volume effects. In such cases, the \(A_{\kappa}\) and \(\tilde{A}_{\kappa}\) modes may acquire non-zero effective masses, due to such finite volume effects, loosing their property of being massless [26; 27; 52].