## 3 Entanglement in the Dissipative Quantum Model of Brain  
In the dissipative quantum model, brain states are considered to be finite temperature states with irreversible time evolution. Thermo field dynamics (TFD) and the irreversible time evolution formalism in QFT [5; 6; 7; 26] require the \"doubling\" of the system degrees of freedom \(A_{k}\): \(A_{\kappa}\to A_{\kappa}\times\tilde{A}_{\kappa}\), for any \(\kappa\). \(\tilde{A}_{\kappa}\) describes the thermal bath or the environment in which the system is embedded. The doubling operation has a well-defined algebraic structure [6; 7] (Hopf algebra [7]). It describes the system-environment interaction, and it leads to considering the \(\{A,\tilde{A}\}\) system as a closed system. Thus, it allows the use of the QFT canonical formalism, which is indeed limited to closed systems. We refer to the \(\tilde{A}\) modes as the _Double_. As we will see, also related to the algebraic structure of the doubling operation is the dynamic origin of the system-environment entanglement. At this stage of the modeling, the Hamiltonian of non-interacting DWQ has the form:  
\[H_{0}=\sum_{\kappa}\hbar\Omega_{\kappa}\big{(}A_{\kappa}^{\dagger}A_{\kappa}- \tilde{A}_{\kappa}^{\dagger}\tilde{A}_{\kappa}\big{)}\, \tag{1}\]  
where \(\Omega_{\kappa}\) is the frequency, and the minus sign accounts for the fact that energy fluxes outgoing from the \(A\) system are ingoing fluxes for the \(\tilde{A}\) system, and vice versa. Because of the in/out exchange role between the \(A\) and \(\tilde{A}\) modes, they are, in this specific sense (not as backward causation), \"time-reversed mirror modes\"; we thus say that \(\tilde{A}_{\kappa}\) is the \"time-reversed image\" of \(A_{\kappa}\). Let us denote by \(\{|\mathcal{N}_{A_{\kappa}},\mathcal{N}_{\tilde{A}_{\kappa}}\rangle\}\) the set of simultaneous eigenvectors of the number operators \(\tilde{N}_{A_{\kappa}}\equiv A_{\kappa}^{\dagger}A_{\kappa}\) and \(\tilde{N}_{\tilde{A}_{\kappa}}\equiv\tilde{A}_{\kappa}^{\dagger}\tilde{A}_{\kappa}\), where \(\mathcal{N}_{A_{\kappa}}\) and \(\mathcal{N}_{\tilde{A}_{\kappa}}\) are non-negative integers, and let \(|0\rangle_{0}\equiv|\mathcal{N}_{A_{\kappa}}=0,\mathcal{N}_{\tilde{A}_{\kappa }}=0\rangle\) be the vacuum annihilated by \(A_{\kappa}\) and \(\tilde{A}_{\kappa}\): \(A_{\kappa}|0\rangle_{0}=0=\tilde{A}_{\kappa}|0\rangle_{0}\) for any \(\kappa\). The eigenvalue of \(H_{0}\) on the state \(|0\rangle_{0}\) and on any state that is the condensate of equal number of \(A_{\kappa}\) and \(\tilde{A}_{\kappa}\) for any \(\kappa\) is thus zero, which expresses the balance of in/out energy fluxes and, therefore, that the couple \(\{A,\tilde{A}\}\) is a closed system. This also means that anyone of these states with an equal number of tilde and non-tilde modes may be taken as a ground state. Consider one of the infinitely many zero-energy states at some initial time \(t_{0}=0\). The memory state is defined [26] to be \(|0\rangle_{\mathcal{N}}\), with \(\mathcal{N}\equiv\{\mathcal{N}_{A_{\kappa}}=\mathcal{N}_{\tilde{A}_{\kappa}}, \forall\kappa,at\ \ t_{0}=0\}\) the set of integers defining the \"initial value\" of the condensate. \(\mathcal{N}\) is thus the \"code\" associated with the information recorded at time \(t_{0}=0\). It turns out [26] that \(|0\rangle_{\mathcal{N}}\) is the two-mode \(SU(1,1)\) generalized squeezed coherent state:  
\[|0\rangle_{\mathcal{N}}=\prod_{\kappa}\ \frac{1}{\cosh\theta_{\kappa}}\ \exp\Big{(}-\tanh\theta_{\kappa}A_{\kappa}^{\dagger}\tilde{A}_{\kappa}^{ \dagger}\Big{)}\ \ |0\rangle_{0}\, \tag{2}\]  
which is normalized to one, \(\mathcal{N}\langle 0|0\rangle_{\mathcal{N}}=1\) for all \(\mathcal{N}\) and where the parameter \(\theta_{\kappa}\) is actually temperature dependent, \(\theta_{\kappa}=\theta_{\kappa}(\beta)\). For notational simplicity, we will avoid to use the temperature dependence and simply write \(\theta_{\kappa}\), for any \(\kappa\).