## 3 Entanglement in the Dissipative Quantum Model of Brain
Then, the non-unitary relation among different representations may be lost, and disentanglement processes may be activated by some unitary operator. This particular feature of the model may be related to some pathological behavior, for example with depression states arising from loss of interest in (disentanglement from) the surrounding world [27; 52]. However, we will not discuss these problems, which are out of the scope of our presentation. We also remark that the memory code \(\mathcal{N}\) is a macroscopic observable in the sense mentioned above, namely it is not affected by quantum fluctuations as a result of the coherence of the DWQ condensation. We have thus the \"change of scale\", from micro to macro: the memory state \(|0\rangle_{\mathcal{N}}\) is a \"macroscopic quantum state\". We may then define the \"brain (ground) state\" as the full set of entangled memory states \(|0\rangle_{\mathcal{N}}\), for all \(\mathcal{N}\). We recall that the \(\tilde{A}_{\kappa}\) modes are the brain \"imaging\" of the environment, and the ordered patterns in the condensate of the \(\{A,\tilde{A}\}\) couples facilitate the formation of neuronal AM synchronized assemblies. As such, the \(\tilde{A}_{\kappa}\) modes have been related [29; 53; 54] to the mental activity of the brain, the \"mind\". The \(A-\tilde{A}\) entanglement thus results in the brain and mind activity entanglement. We will comment further on this point in the following (cf. Section 5). Let us now consider the time evolution of the memory states. The Hamiltonian is [26]:  
\[H=H_{0}+H_{I}=\sum_{\kappa}\hbar\Omega_{\kappa}\big{(}A_{\kappa}^{\dagger}A_{ \kappa}-\tilde{A}_{\kappa}^{\dagger}\tilde{A}_{\kappa}\big{)}+i\sum_{\kappa} \hbar\Gamma_{\kappa}\big{(}A_{\kappa}^{\dagger}\tilde{A}_{\kappa}^{\dagger}-A_{ \kappa}\tilde{A}_{\kappa}\big{)}. \tag{6}\]  
\(\Gamma_{\kappa}\) is the damping constant, and the \(H_{I}\) term accounts for the dissipative time evolution. We immediately remark that the number \((\mathcal{N}_{\Lambda_{\kappa}}-\mathcal{N}_{\tilde{A}_{\kappa}})\) is a constant of motion for any \(\kappa\), since \(H_{0}\) commutes with \(H_{I}\), \([H_{0},H_{I}]=0\). \(H_{0}\) is the Casimir operator of the \(SU(1,1)\) group structure of \(H\). We find [26] that the memory state \(|0\rangle_{\mathcal{N}}\) evolves in time as:  
\[|0(t)\rangle_{\mathcal{N}}=\exp\left(-it\frac{H}{\hbar}\right)|0\rangle_{ \mathcal{N}}=\prod_{\kappa}\frac{1}{\cosh\left(\Gamma_{\kappa}t-\theta_{\kappa }\right)}\exp\left(\tanh\left(\Gamma_{\kappa}t-\theta_{\kappa}\right)A_{k}^{ \dagger}\tilde{A}_{k}^{\dagger}\right)|0\rangle_{0}\, \tag{7}\]  
and:  
\[\mathcal{N}_{\Lambda_{\kappa}}(\theta,t)={}_{\mathcal{N}}\langle 0(t)|A_{ \kappa}^{\dagger}A_{\kappa}|0(t)\rangle_{\mathcal{N}}=\sinh^{2}\!\left(\Gamma _{\kappa}t-\theta_{\kappa}\right)\, \tag{8}\]  
and a similar expression is obtained for the \(\tilde{A}_{\kappa}\) modes. Again, \(|0(t)\rangle_{\mathcal{N}}\) is a normalized \(SU(1,1)\) generalized coherent state, \({}_{\mathcal{N}}\langle 0(t)|0(t)\rangle_{\mathcal{N}}=1\), and it is an entangled state. Thus, all previous conclusions for the state \(|0\rangle_{\mathcal{N}}\) at the initial time extend to the time-evolved states \(|0(t)\rangle_{\mathcal{N}}\) at any \(t\). We remark that since the condensate of the modes \(A\) and \(\tilde{A}\) promotes the synaptic and ephaptic neuronal communication, the entanglement between \(A\) and \(\tilde{A}\) produces neuronal synchronicity. Neurons, which in the model together with other cells are considered to be classical units, embedded in the medium of DWQ act in some sense as \"observers\" (myriads of [PERSON] and [PERSON]) of the collective background of entangled \(A\) and \(\tilde{A}\) modes. Phase-mediated correlations for the couples \((A_{\kappa},\tilde{A}_{\kappa})\) and their coherent dynamics are thus \"transferred\" to neuronal populations.