## 2 Methodology
### Quantum Entanglement  
In the explanation of quantum mechanical phenomena associated with the light harvesting complexes, the accurate characterization of the energy of one system cannot be described without taking into account the other energy states. The key phenomenon of this type is known as quantum entanglement and is particularly important to be considered for the chlorophyll pigments or chromophores present in the reaction centers of light harvesting complex [88; 89; 90]. The formulation of the quantum entanglement for the FMO complex as described here has been adopted from [88]--whereby the state in the single-excitation manifold of the light harvesting complex can be expressed as [88]  
\[\rho(t)=\sum_{i=1}^{N}\rho_{ii}(t)|i\rangle\langle i|+\sum_{i=1}^{N}\sum_{j>i }^{N}\rho_{ij}(t)|i\rangle\langle j|+\rho_{ij}^{*}(t)|j\rangle\langle i|, \tag{9}\]  
where \(\langle i|\) represents the state for which only the \(ith\) chromophore (site) is excited and all other chromophores are in their electronic ground states and \(\rho\) is an unnormalized density matrix of \(N\) chromophores. Furthermore, the mixed-state entanglement in the physically relevant zero- and single-excitation subspaces of the light harvesting complexes can be readily computed from [88]:  
\[E[\rho]=-\sum_{i=1}^{N}\rho_{ii}\ln\rho_{ii}-S(\rho), \tag{10}\]  
where the quantity \(S(\rho)\) is known as the von Neumann entropy of the state \(\rho\) and is given by the relation: \(S(\rho)=-\mathrm{tr}(\rho\ \ln\rho)\)[91]. Moreover, the analysis of physical structure and energy levels of FMO complex indicates that sites 1 and 6 interface with the chlorosome, transmitting energy to the complex, and experiments have also confirmed that sites 1 and 6 are the first to become excited in the FMO protein [3; 88]. The global quantum coherence in the FMO protein at two different temperatures, viz., 77 K (commonly employed temperature for the ultrafast spectroscopy experiments in probing the light harvesting complexes) and physiologically relevant temperature of 300 K, has been presented in Figure 2. This study demonstrates the direct implementation of Equation (10) for the initial state excitation on site \(|1\rangle\) or \(|6\rangle\) (see further details in [88]). As depicted in Figure 2, the general feature of the global entanglement measure for all scenarios is that its value rises rapidly for short times due to the quick delocalization of the excitation caused by large 1-2 and 5-6 site coupling terms in the Hamiltonian of FMO complex and then decays after approximately 30-50 fs with varying amounts of oscillation as the excitation begins to explore other sites owing to incoherent transport and rapid dephasing. Furthermore, there is finite entanglement in the system up to 5 ps for both initial conditions and temperatures. In reality, the quantitative value of the unnormalized density matrix \(\rho\) determines the amount of entanglement in the respective state. This quantum entanglement observed between the different sites or the chromophores or the BChls in the FMO complex might explain the existence of correlations between them that can be clarified utilizing the quantum mechanics approach, which has become a very important tool for analyzing a range of phenomena observed in biological systems. The quantity \(\rho\) (the density matrix of the system in the basis of local excited states given in Equation (9) determines whether two states generating this density matrix are separable or not. Furthermore, the quantum coherence observed between the different BChls present in the FMO light harvesting complex also reshuffles the energy distribution that will initiate the energy transfer among them [92]. It is noteworthy to mention that the importance of the quantum coherence in this context continues to be debated [74; 75; 76; 93; 94; 95; 96; 97]. Figure 2: (Color online) Time evolution of the global quantum entanglement for the two initial states \(|1\rangle\) and \(|6\rangle\) at low (77 K) and high (300 K) temperatures for the FMO protein.