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==11. Compound biosystems==

===11.1. Entanglement of information states of biosystems===

The state space <math>{\mathcal{H}}</math> of the biosystem <math>S</math> consisting of the subsystems <math>S_j,j=1,2,....n</math>, is the tensor product of subsystems’ state spaces<math>{\mathcal{H}}_j</math> , so

The easiest way to imagine this state space is to consider its coordinate representation with respect to some basis constructed with bases in <math>{\mathcal{H}}_j</math>. For simplicity, consider the case of qubit state spaces <math>{\mathcal{H}}_j</math> let <math>|\alpha\rangle</math>, <math>|\alpha\rangle=0,1</math>, be some orthonormal basis in <math>{\mathcal{H}}_j</math>, i.e., elements of this space are linear combinations of the form <math>|\psi_{j}\rangle=c_0|0\rangle+c_1|1\rangle</math>. (To be completely formal, we have to label basis vectors with the index <math>j</math>, i.e.,<math>|\alpha\rangle\equiv |\alpha\rangle_j</math>. But we shall omit this it.) Then vectors  <math>|\alpha_1.....\alpha_n\rangle \equiv |\alpha_1\rangle\otimes....\otimes|\alpha_n\rangle</math> form the orthonormal basis in <math>{\mathcal{H}}</math>, i.e., any state  <math>|{\mathcal{\Psi}}\in {\mathcal{H}} </math> can be represented in the form

and the complex coordinates  <math> C_{\alpha_1.....\alpha_n}</math> are normalized: <math> \sum |C_{\alpha_1.....\alpha_n}|^2=1</math>. For example, if <math> n=2</math>, we can consider the state

This is an example of an ''entangled state'', i.e., a state that cannot be factorized in the tensor product of the states of the subsystems. An example of a non-entangled state (up to normalization) is given by

Entangled states are basic states for quantum computing that explores state’s inseparability. Acting to one concrete qubit modifies the whole state. For a separable state, by transforming say the first qubit, we change only the state of system <math> S_1</math> . This possibility to change the very complex state of a compound system via change of the local state of a subsystem is considered as the root of superiority of quantum computation over classical one. We remark that the dimension of the tensor product state space is very big, it equals <math> 2^n</math> for <math> n</math> qubit subsystems. In quantum physics, this possibility to manipulate with the compound state (that can have the big dimension) is typically associated with “quantum nonlocality” and ''spooky action at a distance.''But, even in quantum physics this nonlocal interpretation is the source for permanent debates []. In particular, in the recent series of papers [] it was shown that it is possible to proceed without referring to quantum nonlocality and that quantum mechanics can be interpreted as the local physical theory. ''The local viewpoint on the quantum theory is more natural for biological application.''13 For biosystems, spooky action at a distance is really mysterious; for humans, it corresponds to acceptance of parapsychological phenomena.

How can one explain generation of state-transformation of the compound system <math> S</math> by “local transformation” of say the state of its subsystem <math> S_1</math>? Here the key-role is played by ''correlations'' that are symbolically encoded in entangled states. For example, consider the compound system <math> S=(S_1,S_2)</math> in the state <math> |\Psi\rangle</math> given by (33). Consider the projection-type observables <math> A_j</math> on <math> A_j</math>represented by Hermitian operators <math>\widehat{A}_j</math> with eigen-vectors <math>|0\rangle</math>,<math>|1\rangle</math> (in qubit spaces <math>{\mathcal{H}}_j</math>). Measurement of say <math> A_1</math> with the output <math> A_1=\alpha</math> induces the state projection onto the vector <math>|\alpha\alpha\rangle</math>.   

Hence, measurement of  <math> A_2</math> will automatically produce the output <math> A_2=\alpha</math>. Thus, the state  <math> |\Psi\rangle</math> encodes the exact correlations for these two observables. In the same way, the state

encodes correlations <math> A_1=\alpha</math>, <math> A_2=\alpha</math> (mod 2).  

Theory of open quantum systems operates with mixed states described by density operators. And before to turn to modeling of biological functions for compound systems, we define entanglement for mixed states. Consider the case of tensor product of two Hilbert spaces, i.e., the system <math> S</math> is compound of two subsystems <math> S_1</math> and <math> S_2</math>. A mixed state of <math> S</math> given by <math>\widehat{\rho}</math> is called separable if it can be represented as a convex combination of product states <math>\widehat{\rho}=\sum_k p_k\widehat{\rho}_{1k}\otimes \widehat{\rho}_{2k}</math>, where <math>\widehat{\rho}ik</math>, <math>i=1,2</math>, are the density operator of the subsystem <math> S_i</math> of <math> S</math>. Non-separable states are called entangled. They symbolically represent correlations between subsystems.

Quantum dynamics describes the evolution of these correlations. In the framework of open system dynamics, a biological function approaches the steady state via the process of decoherence. As was discussed in Section 8.3, this dynamics resolves uncertainty that was initially present in the state of a biosystem; at the same time, it also washes out the correlations: the steady state which is diagonal in the basis <math> \{|\alpha_1....\alpha_2\rangle\}</math> is separable (disentagled). However, in the process of the state-evolution correlations between subsystems (entanglement) play the crucial role. Their presence leads to transformations of the state of the compound system  <math> S</math> via “local transformations” of the states of its subsystems. Such correlated dynamics of the global information state reflects ''consistency of the transformations of the states of subsystems.''

Since the quantum-like approach is based on the quantum information representation of systems’ states, we can forget about the physical space location of biosystems and work in the information space given by complex Hilbert space <math>{\mathcal{H}}</math>. In this space, we can introduce the notion of locality based on the fixed tensor product decomposition (31). Operations in its components <math>{\mathcal{H}}_j</math> we can call local (in information space). But, they induce “informationally nonlocal” evolution of the state of the compound system.

===11.2. Entanglement of genes’ epimutations===

Now, we come back to the model presented in Section 9 and consider the information state of cell’s epigenome expressing potential epimutations of the chromatin-marking type. Let cell’s genome consists of <math>m</math> genes <math>g_1,....,g_m</math>. For each gene <math>g</math>, consider all its possible epimutations and enumerate them: <math>j_g=1,......k_g</math>. The state of all potential epimutations in the gene <math>g</math> is represented as superposition

In the ideal situation – epimutations of the genes are independent – the state of cell’s epigenome is mathematically described by the tensor product of the states <math>|\psi_g\rangle</math>:  

However, in a living biosystem, the most of the genes and proteins are correlated forming a big network system. Therefore, one epimutation affects other genes. In the quantum information framework, this situation is described by entangled states:

This form of representation of potential epimutations in the genome of a cell implies that epimutation in one gene is consistent with epimutations in other genes. If the state is entangled (not factorized), then by acting, i.e., through change in the environment, to one gene, say <math>g_1</math>, and inducing some epimutation in it, the cell “can induce” consistent epimutations in other genes.

Linearity of the quantum information representation of the biophysical processes in a cell induces the linear state dynamics. This makes the epigenetic evolution very rapid; the off-diagonal elements of the density matrix decrease exponentially quickly. Thus, our quantum-like model justifies the high speed of the epigenetic evolution. If it were based solely on the biophysical representation with nonlinear state dynamics, it would be essentially slower.

Modeling based on theory of open systems leads to reconsideration of interrelation between the Darwinian with Lamarckian viewpoint on evolution. Here we concentrated on epimutations, but in the same way we can model mutations (Asano et al., 2015b).