Store:QLMde15

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 of nonzero 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.