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8. Open quantum systems: interaction of a biosystem with its environment

As was already emphasized, any biosystem  is fundamentally open. Hence, dynamics of its state has to be modeled via an interaction with surrounding environment . The states of   and are represented in the Hilbert spaces  and . The compound system  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:

 

where  is the pure state of the system  and  is its Hamiltonian. This equation implies that the pure state  evolves unitarily :. Here . Hamiltonian (evolution-generator) describing information interactions has the form , where  ,are Hamiltonians of the systems and  is the interaction Hamiltonian.12 This equation implies that evolution of the density operator  of the system  is described by von Neumann equation:

 

However, the state   is too complex for any mathematical analysis: the environment includes too many degrees of freedom. Therefore, we are interested only the state of ; its dynamics is obtained via tracing of the state of   w.r.t. the degrees of freedom of  :

 

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):

 

where Hermitian operator (Hamiltonian)  describes the internal dynamics of  and the superoperator , acting in the space of density operators, describes an interaction with environment . 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.