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==8. Open quantum systems: interaction of a biosystem with its environment==
As was already emphasized, any biosystem <math>S</math> is fundamentally open. Hence, dynamics of its state has to be modeled via an interaction with surrounding environment <math>
\varepsilon</math>. The states of  <math>S</math> and <math>
\varepsilon</math> are represented in the Hilbert spaces <math>\mathcal{H}</math> and <math>\mathcal{H}</math>. The compound system <math>S+\varepsilon</math> is represented in the tensor product Hilbert spaces . This system is treated as an isolated system and in accordance with quantum theory, dynamics of its pure state can be described by the Schrödinger equation: 

{| width="80%" |
|-
| width="33%" |&nbsp;
| width="33%" |<math>i\tfrac{d}{dt}\Psi(t)=\widehat{H}\Psi(t)(t), \Psi(0)=\Psi_0</math>
| width="33%" align="right" |<math>(21)</math>
|}

where <math>\psi(t)</math> is the pure state of the system <math>S+\varepsilon</math> and <math>\hat{\mathcal{H}}</math> is its Hamiltonian. This equation implies that the pure state <math>\psi(t)</math> evolves unitarily :<math>\psi(t)=\hat{U}(t)\psi_0</math>. Here <math>\hat{U}(t)=e^{-it\hat{\mathcal{H}}}</math>. Hamiltonian (evolution-generator) describing information interactions has the form <math>\hat{\mathcal{H}}=\hat{\mathcal{H}}_s+\hat{\mathcal{H}}_\varepsilon+{\mathcal{\hat H_{S,\varepsilon}}}</math>, where  <math>\hat{\mathcal{H}}_s</math>,<math>\hat{\mathcal{H}}_\varepsilon</math>are Hamiltonians of the systems and  <math>{\mathcal{\hat H_{S,\varepsilon}}}</math>is the interaction Hamiltonian.12 This equation implies that evolution of the density operator <math>\hat{\mathcal{R}}(t)</math> of the system <math>S+\varepsilon</math> is described by von Neumann equation: 

{| width="80%" |
|-
| width="33%" |&nbsp;
| width="33%" |<math>\tfrac{d\widehat{R}}{dt}(t)=-i[\widehat{H},\widehat{R},(t)], \widehat{R}(0)=\widehat{R}_0</math>
| width="33%" align="right" |<math>(22)</math>
|}

However, the state  <math>\hat{\mathcal{R}}(t)</math> is too complex for any mathematical analysis: the environment includes too many degrees of freedom. Therefore, we are interested only the state of <math>S</math>; its dynamics is obtained via tracing of the state of  <math>S+\varepsilon</math> w.r.t. the degrees of freedom of <math>\varepsilon</math> :

{| width="80%" |
|-
| width="33%" |&nbsp;
| width="33%" |<math>\widehat{\rho}(t)=Tr_\mathcal{H}\widehat{R}(t)</math>
| width="33%" align="right" |<math>(23)</math>
|}

Generally this equation, ''the quantum master equation'', is mathematically very complicated. A variety of approximations is used in applications.

===8.1. Quantum Markov model: Gorini–Kossakowski–Sudarshan–Lindbladequation===
The simplest approximation of quantum master equation (23) is ''the quantum Markov dynamics'' given by the ''Gorini–Kossakowski–Sudarshan–Lindblad'' (GKSL) equation (Ingarden et al., 1997) (in physics, it is commonly called simply the Lindblad equation; this is the simplest quantum master equation): 

{| width="80%" |
|-
| width="33%" |&nbsp;
| width="33%" |<math>\tfrac{d\widehat{\rho}}{dt}(t)=-i[\widehat{H},\widehat{\rho},(t)]+ \widehat{L}[\widehat{\rho}(t),\widehat{\rho}(0)=\widehat{\rho}_0</math>
| width="33%" align="right" |<math>(24)</math>
|}

where Hermitian operator (Hamiltonian) <math>\widehat{\mathcal{H}}</math> describes the internal dynamics of <math>S</math> and the superoperator <math>\widehat{{L}}</math>, acting in the space of density operators, describes an interaction with environment <math>\varepsilon</math>. This superoperator is often called ''Lindbladian.'' The GKSL-equation is a quantum master equation for Markovian dynamics. In this paper, we have no possibility to explain the notion of quantum Markovianity in more detail. Quantum master equation (23) describes generally non-Markovean dynamics.
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