## 3 Quantum Modelling of FMO Quantum Excitation
### Entanglement Generation Depending of \(T,\lambda_{k}\) and \(\gamma_{k}\)  
Each FMO dimmer has components interacting among them, transmitting the initial excitation in the antennas (BChl 8 for the model with seven BChls, BChl 1 or 6 for the original model with seven BChls) to the remaining BChls through multipartite interactions. This dynamic generates complex entanglement among BChls [63]. Despite the limitation to quantify it, we can still address this phenomenon by using the partial trace criteria to get the concurrence as a measure of entanglement between different parts [61]. First, we trace the whole system except for two BChl states \(k<l\), and then all subsystems except the \(k\)-th BChl to get, respectively:  
\[\rho_{\{kl\}} = \text{Tr}_{\{kl\}}(\rho)=\sum_{i,j=1}^{N}\rho_{ij}|\delta_{ki} \delta_{li}\rangle\langle\delta_{kj}\delta_{lj}|\prod_{k,l\
eq s=1}^{N} \langle\delta_{si}|\delta_{sj}\rangle \tag{33}\] \[= \sum_{i\
eq k,l}\rho_{ii}|00\rangle\langle 00|+\rho_{kk}|10 \rangle\langle 10|+\rho_{ll}|01\rangle\langle 01|+\rho_{kl}|10\rangle\langle 01|+ \rho_{lk}|01\rangle\langle 10|\] \[\rho_{\{k\}} = \text{Tr}_{\{k\}}(\rho)=(1-\rho_{kk})|0\rangle\langle 0|+\rho_{kk} |1\rangle\langle 1 \tag{34}\]  
Then, taking the concurrence definitions for each kind of expression: the first for a two system density matrix, and the second for one system related with the remainder [58; 61], we get:  
\[\mathcal{C}_{\{kl\}} \equiv \mathcal{C}(\rho_{\{kl\}})=2|\rho_{kl}| \tag{35}\] \[\mathcal{C}_{\{k\}} \equiv \mathcal{C}(\rho_{\{k\}})=2\sqrt{\rho_{kk}(1-\rho_{kk})} \tag{36}\]  
which can be respectively interpreted as the entanglement (a) among systems \(k\) and \(l\)[43], and (b) between the system \(k\) and the remaining BChls [58]. Each concurrence ranges from zero (separable) to one (maximally entangled). Figure 11 is a compact graphical comparative representation of the entanglement dynamics in the excitation process through the previous concurrences present between each BChl and the rest of the system, as well as each one with each other at the fixed \(\gamma=50\,\text{cm}^{-1}\) and (A) \(\lambda=35\,\text{cm}^{-1}\) or (B) \(\lambda=65\,\text{cm}^{-1}\) as a function of \(t\) and \(T\). These values are obtained from the application of (36) with the density matrices (\(\rho_{kk}\)) generated by the HEOM calculations previously depicted. The graphs show the time evolution and temperature dependence of the entanglement through a color gradient in the scale 0 (red) to 1 (blue) indicated in the color bar above. Each circular sector of the plot represents the concurrence experienced by each BChl molecule in the FMO complex. The central circular region represents \(C_{\{k\}}\) for each BChl indicated by numbers around. While, the following ring sectors represent \(C_{\{kl\}}\) pairing the numbers around with those for each track on the right side. In each region, time \(t\) increases in counterclockwise direction from \(0\,\text{ps}\) to \(2\,\text{ps}\) repetitively inside each track (indicated by the blue arrow), while radially outwards each track has a temperature gradient from \(77\,^{\circ}\)K to \(347\,^{\circ}\)K (indicated by the brown arrow). It is important to note that at the beginning (as well as final states in the equilibrium) of the transfer process, all BChls are separable states. It is only during a brief period of time that entanglement appears. For that reason, only the first \(2\,\text{ps}\) are presented in Figure 11. For \(C_{\{k\}}\) in the center, we note that the main BChl entangled with the remainder BChls is BChl 1 followed by BChl 2, and then by BChl 3 and BChl 4 with a delay. Other BChls have a lower global entanglement. It is notorious that the highest entanglement readings are usually between BChl 1 and the rest of the system, but particularly with BChl 2 and 3 as seen in the corresponding rings of both Figure 11A,B. Of further interest is the increasing in entanglement between BChls 3 and 4, 4 and 5, and 4 and 7 as time increases.