## 3 Quantum Modelling of FMO Quantum Excitation
### Coherence Depending of \(T\), \(\lambda_{k}\) and \(\gamma_{k}\)  
BChls in FMO complexes exhibit an initial increase and sustained quantum coherence, so we are interested to define quantitatively such an aspect. A quantum state is said coherent if it exhibits the form of a pure state: \(|\psi\rangle=\sum_{i}\alpha_{i}|\phi_{i}\rangle\). Still, in terms of the density matrix, \(\rho=|\psi\rangle\langle\psi|=\sum_{i,j}\rho_{ij}|\phi_{i}\rangle\langle\phi_ {j}|\), such state is said non-coherent if \(\rho_{ij}=\rho_{ii}\delta_{ij}\) with \(\sum_{i}\rho_{ii}=1\) and \(\rho_{ii}\
eq 0\) for at least a couple of values of \(i\). It means that a state exhibit coherence if at least one of the \(\rho_{ij}\
eq 0,i\
eq j\). Then, a coherence measure should fulfill certain requirements due to the quantum nature of such a property. If \(C(\rho)\) is a coherence measure, then it should fulfill [59]: (a) Non-negativity, \(C(\rho)\geq 0\) and \(C(\rho)=0\iff\rho\) is non-coherent. (b) If \(\Lambda_{i}\) is a completely positive trace preserving map (CPTP) lowering the coherence, then \(C(\Lambda_{i}[\rho])\leq C[\rho]\). c) \(C(\sum_{i}p_{i}\rho_{i})\leq\sum_{i}p_{i}C(\rho_{i})\), which means \(C\) is convex, the mixture of states has less coherence than the average of coherence. There are several candidates  
Figure 8: Contour plots for the equilibrium populations \(\rho_{ii_{\infty}}\) behavior in color for each BChl as function of \(T\),\(\lambda_{k}\),\(\gamma_{k}\). (**A–C**) those with the largest variability, and (**D–G**) those with the lowest. Insets are alternative equivalent plots in transparent colors in the analysis region. Curves on the contours are cuts with constant \(T\) (green), \(\lambda_{k}\) (red) and \(\gamma_{k}\) (yellow). for such a measure. If the set of the non-coherent states is \(\Gamma\), the first one is a measure based on the minimum distance from the any non-coherent state by using the \(l_{1}\)-norm [60]:  
\[C_{l_{1}}(\rho)=\min_{\sigma\in\Gamma}\sum_{ij}|\rho_{ij}-\sigma_{ij}|=2\sum_{i< j}|\rho_{ij}| \tag{31}\]  
for \(\sigma_{\text{diag}}=\sum_{i}\rho_{ii}|\phi_{i}\rangle\langle\phi_{i}|\) having the minimum. \(C_{l_{1}}(\rho)\leq d-1\) for qudits is bounded by the maximally coherent state \(|\psi_{mc}\rangle=\frac{1}{\sqrt{d}}\sum_{i}|\phi_{i}\rangle\). Another measure, the relative entropy of coherence [59], defined by the relative \(S(\rho\|\sigma)\) and the von Newmann entropies \(S(\rho)\)[61]:  
\[C_{re}(\rho)=\min_{\sigma\in\Gamma}S(\rho\|\sigma)=S(\rho_{\text{diag}})-S(\rho) \tag{32}\]  
measuring the minimum amount of noise to destroy the coherence remaining in the system. The minimum is reached again by \(\sigma=\rho_{\text{diag}}\). It is bounded \(0\leq C_{re}(\rho)\leq C_{l_{1}}(\rho)\leq d-1\) when \(S(\rho)\) is defined using \(\ln(\rho)\) instead \(\log_{2}(\rho)\)[62]. Despite \(C_{re}(\rho)\) fulfills the entire previous properties for a desired coherence measurement [59], in our analysis, we use \(C_{l_{1}}(\rho)\) for simplicity (despite last property is not fulfilled by it [59], but alternatively this quantity is nearly related with one of the entanglement measure used for such a purpose also, as we will be seen). For our parametric analysis, Figure 9A is a graphical representation of the parametric dependence of the system's maximum coherence. For the original data obtained from the simulations, the entire general range for \(C_{l_{1}\max}(\rho)\) in the region became \([1.22,2.33]\). As before, departing from the group of parametric simulations, we are calculated for each one the maximum value of \(C_{l_{1}}(\rho)\) reached and then approximated through the method of maximum likelihood (with \(\Delta^{2}_{C_{l_{1}\max}(\rho)}=4.7\times 10^{-3}\)). Then, some contours for constant values of \(C_{l_{1}\max}(\rho)\) in agreement with the color scale below (note the maximum mathematically possible value is \(d-1=6\)).