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===8.2. Biological functions in the quantum Markov framework===
We turn to the open system dynamics with the GKSL-equation. In our modeling, Hamiltonian  <math>\widehat{\mathcal{H}}</math> and Lindbladian  <math>\widehat{{L}}</math> represent some special ''biological function'' <math>F</math> (see Khrennikov et al., 2018) for details. Its functioning results from interaction of internal and external information flows. In Sections 10, 11.3,  <math>F</math> is some ''psychological function''; in the simplest case <math>F</math> represents a question asked to  <math>S</math> (say  is a human being). In Section 7, <math>F</math>  is the ''gene regulation'' of glucose/lactose metabolism in Escherichia coli bacterium. In Sections 9, 11.2,  <math>F</math> represents the process of ''epigenetic mutation''. Symbolically biological function <math>F</math> is represented as a quantum observable: Hermitian operator  <math>\widehat{F}</math> with the spectral decomposition <math>\widehat{F}=\sum_xx\widehat{E}^F(x)</math>, where <math>x</math> labels outputs of <math>F</math>. Theory of quantum Markov state-dynamics describes the process of generation of these outputs.

In the mathematical model (Asano et al., 2015b, Asano et al., 2017b, Asano et al., 2017a, Asano et al., 2015a, Asano et al., 2012b, Asano et al., 2011, Asano et al., 2012a), the outputs of biological function <math>F</math>  are generated via approaching a ''steady state'' of the GKSL-dynamics: 

{| width="80%" |
|-
| width="33%" |&nbsp;
| width="33%" |<math>\lim_{t \to \infty}\widehat{\rho}(t)=\widehat{\rho}_{steady}</math>
| width="33%" align="right" |<math>(25)</math>
|}

such that it matches the spectral decomposition of <math>\widehat{F}</math>, i.e.,  

{| width="80%" |
|-
| width="33%" |&nbsp;
| width="33%" |<math>\widehat{\rho}_{steady}=\sum_x p_x\widehat{E}^F(x)</math> 
| width="33%" align="right" |<math>(26)</math>
|}

where

{| width="80%" |
|-
| width="33%" |&nbsp;
| width="33%" |<math>p_x\geq\sum_xp_x=1</math>
| width="33%" align="right" |<math>(26)</math>
|}

This means that <math>\widehat{\rho}_{steady}</math> is diagonal in an orthonormal basis consisting of eigenvectors of <math>\widehat{F}</math>. This state, or more precisely, this decomposition of density operator <math>\widehat{\rho}_{steady}</math>, is the classical statistical mixture of the basic information states determining this biological function. The probabilities in state’s decomposition (26) are interpreted statistically.

Consider a large ensemble of biosystems with the state <math>\widehat{\rho}_0</math> interacting with environment <math>\varepsilon</math>. (We recall that mathematically the interaction is encoded in the Lindbladian <math>\widehat{{L}}</math>) Resulting from this interaction, biological function <math>F</math> produces output <math>x</math> with probability <math>p_x</math>. We remark that in the operator terms the probability is expressed as <math>p_x=Tr\widehat{\rho}_{steady}\widehat{E}^F(x)</math>

This interpretation can be applied even to a single biosystem that meets the same environment many times.

It should be noted that limiting state  <math>\widehat{\rho}_{steady}</math> expresses the stability with respect to the influence of concrete environment <math>\varepsilon</math>. Of course, in the real world the limit-state would be never approached. The mathematical formula (25) describes the process of stabilization, damping of fluctuations. But, they would be never disappear completely with time.

We note that a steady state satisfies the stationary GKSL-equation: 

{| width="80%" |
|-
| width="33%" |&nbsp;
| width="33%" |<math>i[\widehat{H},\widehat{\rho}_{steady}]= \widehat{L}[\widehat{\rho}_{steady}]</math>
| width="33%" align="right" |<math>(27)</math>
|}

It is also important to point that generally a steady state of the quantum master equation is not unique, it depends on the class of initial conditions.