Store:Khrennikov13

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

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

$i{\tfrac {d}{dt}}\Psi (t)={\widehat {H}}\Psi (t)(t),\Psi (0)=\Psi _{0}$

$(21)$

where $\psi (t)$ is the pure state of the system $S+\varepsilon$ and ${\hat {\mathcal {H}}}$ is its Hamiltonian. This equation implies that the pure state $\psi (t)$ evolves unitarily :$\psi (t)={\hat {U}}(t)\psi _{0}$. Here ${\hat {U}}(t)=e^{-it{\hat {\mathcal {H}}}}$. Hamiltonian (evolution-generator) describing information interactions has the form ${\hat {\mathcal {H}}}={\hat {\mathcal {H}}}_{s}+{\hat {\mathcal {H}}}_{\varepsilon }+{\mathcal {{\hat {H}}_{S,\varepsilon }}}$, where  ${\hat {\mathcal {H}}}_{s}$,${\hat {\mathcal {H}}}_{\varepsilon }$are Hamiltonians of the systems and  ${\mathcal {{\hat {H}}_{S,\varepsilon }}}$is the interaction Hamiltonian.12 This equation implies that evolution of the density operator ${\hat {\mathcal {R}}}(t)$ of the system $S+\varepsilon$ is described by von Neumann equation:

${\tfrac {d{\widehat {R}}}{dt}}(t)=-i[{\widehat {H}},{\widehat {R}},(t)],{\widehat {R}}(0)={\widehat {R}}_{0}$

$(22)$

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

${\widehat {\rho }}(t)=Tr_{\mathcal {H}}{\widehat {R}}(t)$

$(23)$

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):

${\tfrac {d{\widehat {\rho }}}{dt}}(t)=-i[{\widehat {H}},{\widehat {\rho }},(t)]+{\widehat {L}}[{\widehat {\rho }}(t),{\widehat {\rho }}(0)={\widehat {\rho }}_{0}$

$(24)$

where Hermitian operator (Hamiltonian) ${\widehat {\mathcal {H}}}$ describes the internal dynamics of $S$ and the superoperator ${\widehat {L}}$, acting in the space of density operators, describes an interaction with environment $\varepsilon$. 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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8.2. Biological functions in the quantum Markov framework

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

$\lim _{t\to \infty }{\widehat {\rho }}(t)={\widehat {\rho }}_{steady}$

$(25)$

such that it matches the spectral decomposition of ${\widehat {F}}$, i.e.,

${\widehat {\rho }}_{steady}=\sum _{x}p_{x}{\widehat {E}}^{F}(x)$

$(26)$

where

$p_{x}\geq \sum _{x}p_{x}=1$

$(26)$

This means that ${\widehat {\rho }}_{steady}$ is diagonal in an orthonormal basis consisting of eigenvectors of ${\widehat {F}}$. This state, or more precisely, this decomposition of density operator ${\widehat {\rho }}_{steady}$, 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 ${\widehat {\rho }}_{0}$ interacting with environment $\varepsilon$. (We recall that mathematically the interaction is encoded in the Lindbladian ${\widehat {L}}$) Resulting from this interaction, biological function $F$ produces output $x$ with probability $p_{x}$. We remark that in the operator terms the probability is expressed as $p_{x}=Tr{\widehat {\rho }}_{steady}{\widehat {E}}^{F}(x)$

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

It should be noted that limiting state  ${\widehat {\rho }}_{steady}$ expresses the stability with respect to the influence of concrete environment $\varepsilon$. 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:

$i[{\widehat {H}},{\widehat {\rho }}_{steady}]={\widehat {L}}[{\widehat {\rho }}_{steady}]$

$(27)$

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.

8.3. Operation of biological functions through decoherence

To make the previous considerations concrete, let us consider a pure quantum state as the initial state. Suppose that a biological function  $F$ is dichotomous, $F=0,1$, and it is symbolically represented by the Hermitian operator that is diagonal in orthonormal basis $|0\rangle$,$|1\rangle$ . (We consider the two dimensional state space — the qubit space.) Let the initial state has the form of superposition

$\psi \rangle =c_{0}|0\rangle +c_{1}|1\rangle$

$(28)$

where $c_{j}\in C,|c_{0}|^{2}+||c_{1}|^{2}=1$. The quantum master dynamics is not a pure state dynamics: sooner or later (in fact, very soon), this superposition representing a pure state will be transferred into a density matrix representing a mixed state. Therefore, from the very beginning it is useful to represent superposition (28) in terms of a density matrix:

${\widehat {\rho }}_{0}={\begin{vmatrix}|c_{0}|^{2}&c_{0}{\bar {c}}_{1}\\{\bar {c}}_{0}c_{1}&|c_{1}|^{2}\end{vmatrix}}$

$(29)$

State’s purity, superposition, is characterized by the presence ofnonzero off-diagonal terms.

Superposition encodes uncertainty with respect to the concrete state basis, in our case $|0\rangle$,$|1\rangle$. Initially biological function $F$  was in the state of uncertainty between two choices $x=0,1$. This is genuine quantum(-like) uncertainty. Uncertainty, about possible actions in future. For example, for psychological function (Section 10)  $F$ representing answering to some question, say “to buy property” ( $F=1$) and its negation ( $F=0$) , a person whose state is described by superposition (28) is uncertain to act with ( $F=1$)  or with ( $F=0$) . Thus, a superposition-type state describes individual uncertainty, i.e., uncertainty associated with the individual biosystem and not with an ensemble of biosystems; with the single act of functioning of $F$  and not with a large series of such acts.

Resolution of uncertainty with respect to ${\widehat {F}}-basis$ is characterized by washing off the off-diagonal terms in (29) The quantum dynamics (24) suppresses the off-diagonal terms and, finally, a diagonal density matrix representing a steady state of this dynamical systems is generated:

${\widehat {\rho }}_{0}={\begin{vmatrix}p_{0}&0\\0&p_{1}\end{vmatrix}}$

$(30)$

This is a classical statistical mixture. It describes an ensemble of biosystems; statistically they generate outputs $F=\alpha$ with probabilities $p_{\alpha }$. In the same way, the statistical interpretation can be used for a single system that performs $F$-functioning at different instances of time (for a long time series).

In quantum physics, the process of washing off the off-diagonal elements in a density matrix is known as the process of decoherence. Thus, the described model of can be called operation of biological function through decoherence.

8.4. Linearity of quantum representation: exponential speed up for biological functioning

The quantum-like modeling does not claim that biosystems are fundamentally quantum. A more natural picture is that they are a complex classical biophysical systems and the quantum-like model provides the information representation of classical biophysical processes, in genes, proteins, cells, brains. One of the advantages of this representation is its linearity. The quantum state space is a complex Hilbert space and dynamical equations are linear differential equations. For finite dimensional state spaces, these are just ordinary differential equations with complex coefficients (so, the reader should not be afraid of such pathetic names as Schrödinger, von Neumann, or Gorini–Kossakowski–Sudarshan–Lindblad equations). The classical biophysical dynamics beyond the quantum information representation is typically nonlinear and very complicated. The use of the linear space representation simplifies the processing structure. There are two viewpoints on this simplification, external and internal. The first one is simplification of mathematical modeling, i.e., simplification of study of bioprocesses (by us, external observers). The second one is more delicate and interesting. We have already pointed to one important specialty of applications of the quantum theory to biology. Here, systems can perform self-observations. So, in the process of evolution say a cell can “learn” via such self-observations that it is computationally profitable to use the linear quantum-like representation. And now, we come to the main advantage of linearity.

The linear dynamics exponentially speeds up information processing. Solutions of the GKSL-equation can be represented in the form ${\widehat {\rho }}(t)=e^{t{\widehat {\Gamma }}}{\widehat {\rho }}$, where ${\widehat {\Gamma }}$ is the superoperator given by the right-hand side of the GKSL-equation. In the finite dimensional case, decoherence dynamics is expressed via factors of the form $e^{t{(ia-b)}}$, where $b>0$. Such factors are exponentially decreasing. Quantum-like linear realization of biological functions is exponentially rapid comparing with nonlinear classical dynamics.

The use of the quantum information representation means that generally large clusters of classical biophysical states are encoded by a few quantum states. It means huge information compressing. It also implies increasing of stability in state-processing. Noisy nonlinear classical dynamics is mapped to dynamics driven by linear quantum(-like) equation of say GKSL-type.

The latter has essentially simpler structure and via selection of the operator coefficients encoding symbolically interaction within the system  $S$ and with its surrounding environment $\varepsilon$,  $S$ can establish dynamics with stabilization regimes leading to steady states.