Resources:Quantum-like modeling in biology with open quantum systems and instruments

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Quantum-like modeling in biology with open quantum systems and instruments

Free resource by Irina Basieva  · Andrei Khrennikov · Masanao Ozawa

Abstract

We present the novel approach to mathematical modeling of information processes in biosystems. It explores the mathematical formalism and methodology of quantum theory, especially quantum measurement theory. This approach is known as quantum-like and it should be distinguished from study of genuine quantum physical processes in biosystems (quantum biophysics, quantum cognition). It is based on quantum information representation of biosystem’s state and modeling its dynamics in the framework of theory of open quantum systems. This paper starts with the non-physicist friendly presentation of quantum measurement theory, from the original von Neumann formulation to modern theory of quantum instruments. Then, latter is applied to model combinations of cognitive effects and gene regulation of glucose/lactose metabolism in Escherichia coli bacterium. The most general construction of quantum instruments is based on the scheme of indirect measurement, in that measurement apparatus plays the role of the environment for a biosystem. The biological essence of this scheme is illustrated by quantum formalization of Helmholtz sensation–perception theory. Then we move to open systems dynamics and consider quantum master equation, with concentrating on quantum Markov processes. In this framework, we model functioning of biological functions such as psychological functions and epigenetic mutation.

1. Introduction

The standard mathematical methods were originally developed to serve classical physics. The real analysis served as the mathematical basis of Newtonian mechanics (Newton, 1687) (and later Hamiltonian formalism); classical statistical mechanics stimulated the measure-theoretic approach to probability theory, formalized in Kolmogorov’s axiomatics (Kolmogorov, 1933). However, behavior of biological systems differ essentially from behavior of mechanical systems, say rigid bodies, gas molecules, or fluids. Therefore, although the ‘‘classical mathematics’’ still plays the crucial role in biological modeling, it seems that it cannot fully describe the rich complexity of biosystems and peculiarities of their behavior — as compared with mechanical systems. New mathematical methods for modeling biosystems are on demand.

In this paper, we present the applications of the mathematical formalism of quantum mechanics and its methodology to modeling biosystems’ behavior. The recent years were characterized by explosion of interest to applications of quantum theory outside of physics, especially in cognitive psychology, decision making, information processing in the brain, molecular biology, genetics and epigenetics, and evolution theory. We call the corresponding models quantum-like. They are not directed to micro-level modeling of real quantum physical processes in biosystems, say in cells or brains (cf. with biological applications of genuine quantum physical theory Penrose, 1989; Umezawa, 1993; Hameroff, 1994; Vitiello, 1995, 2001; Arndt et al., 2009; Bernroider and Summhammer, 2012; Bernroider, 2017). Quantum-like modeling works from the viewpoint to quantum theory as a measurement theory. This is the original Bohr’s viewpoint that led to the Copenhagen interpretation of quantum mechanics (see Plotnitsky, 2009 for detailed and clear presentation of Bohr’s views). One of the main bio-specialties is consideration of self-measurements that biosystems perform on themselves. In our modeling, the ability to perform self-measurements is considered as the basic feature of biological functions (see Section 8.2 and paper Khrennikov et al., 2018).

Quantum-like models (Khrennikov, 2004b) reflect the features of biological processes that naturally match the quantum formalism. In such modeling, it is useful to explore quantum information theory, which can be applied not just to the micro-world of quantum systems. Generally, systems processing information in the quantum-like manner need not be quantum physical systems; in particular, they can be macroscopic biosystems. Surprisingly, the same mathematical theory can be applied at all biological scales: from proteins, cells and brains to humans and ecosystems; we can speak about quantum information biology (Asano et al., 2015a).

In quantum-like modeling, quantum theory is considered as calculus for prediction and transformation of probabilities. Quantum probability (QP) calculus (Section 2) differs essentially from classical probability (CP) calculus based on Kolmogorov’s axiomatics (Kolmogorov, 1933). In CP, states of random systems are represented by probability measures and observables by random variables; in QP, states of random systems are represented by normalized vectors in a complex Hilbert space (pure states) or generally by density operators (mixed states). Superpositions represented by pure states are used to model uncertainty which is yet unresolved by a measurement. The use of superpositions in biology is illustrated by Fig. 1 (see Section 10 and paper Khrennikov et al., 2018 for the corresponding model). The QP-update resulting from an observation is based on the projection postulate or more general transformations of quantum states — in the framework of theory of quantum instruments (Davies and Lewis, 1970; Davies, 1976; Ozawa, 1984; Yuen, 1987; Ozawa, 1997, 2004; Okamura and Ozawa, 2016) (Section 3).

We stress that quantum-like modeling elevates the role of convenience and simplicity of quantum representation of states and observables. (We pragmatically ignore the problem of interrelation of CP and QP.) In particular, the quantum state space has the linear structure and linear models are simpler. Transition from classical nonlinear dynamics of electrochemical processes in biosystems to quantum linear dynamics essentially speeds up the state-evolution (Section 8.4). However, in this framework ‘‘state’’ is the quantum information state of a biosystem used for processing of special quantum uncertainty (Section 8.2).

In textbooks on quantum mechanics, it is commonly pointed out that the main distinguishing feature of quantum theory is the presence of incompatible observables. We recall that two observables, A and B are incompatible if it is impossible to assign values to them jointly. In the probabilistic model, this leads to impossibility to determine their joint probability distribution (JPD). The basic examples of incompatible observables are position and momentum of a quantum system, or spin (or polarization) projections onto different axes. In the mathematical formalism, incompatibility is described as noncommutativity of Her- ̂mitian operators A ̂ and B ̂ representing observables, i.e., [ A,̂ B] ̂ ≠ 0. Here we refer to the original and still basic and widely used model of quantum observables, Von Neumann (1955) (Section 3.2).

Incompatibility–noncommutativity is widely used in quantum physics and the basic physical observables, as say position and momen- tum, spin and polarization projections, are traditionally represented in this paradigm, by Hermitian operators. We also point to numerous applications of this approach to cognition, psychology, decision mak- ing (Khrennikov, 2004a; Busemeyer and Bruza, 2012; Bagarello, 2019) (see especially article (Bagarello et al., 2018) which is devoted to quan- tification of the Heisenberg uncertainty relations in decision making). Still, it may be not general enough for our purpose — to quantum- like modeling in biology, not any kind of non-classical bio-statistics can be easily delegated to von Neumann model of observations. For example, even very basic cognitive effects cannot be described in a way consistent with the standard observation model (Khrennikov et al., 2014; Basieva and Khrennikov, 2015).

We shall explore more general theory of observations based on quantum instruments (Davies and Lewis, 1970; Davies, 1976; Ozawa, 1984; Yuen, 1987; Ozawa, 1997, 2004; Okamura and Ozawa, 2016) and find useful tools for applications to modeling of cognitive ef- fects (Ozawa and Khrennikov, 2020a,b). We shall discuss this ques- tion in Section 3 and illustrate it with examples from cognition and molecular biology in Sections 6, 7. In the framework of the quantum instrument theory, the crucial point is not commutativity vs. noncom- mutativity of operators symbolically representing observables, but the mathematical form of state’s transformation resulting from the back ac- tion of (self-)observation. In the standard approach, this transformation is given by an orthogonal projection on the subspace of eigenvectors corresponding to observation’s output. This is the projection postulate. In quantum instrument theory, state transformations are more general.

Calculus of quantum instruments is closely coupled with theory of open quantum systems (Ingarden et al., 1997), quantum systems inter- acting with environments. We remark that in some situations, quantum physical systems can be considered as (at least approximately) iso- lated. However, biosystems are fundamentally open. As was stressed by Schrödinger (1944), a completely isolated biosystem is dead. The latter explains why the theory of open quantum systems and, in particular, the quantum instruments calculus play the basic role in applications to biology, as the mathematical apparatus of quantum information biology (Asano et al., 2015a).

Within theory of open quantum systems, we model epigenetic evolu- tion (Asano et al., 2012b, 2015b) (Sections 9, 11.2) and performance of psychological (cognitive) functions realized by the brain (Asano et al., 2011, 2015b; Khrennikov et al., 2018) (Sections 10, 11.3).

For mathematically sufficiently well educated biologists, but with- out knowledge in physics, we can recommend book (Khrennikov, 2016a) combining the presentations of CP and QP with a brief intro- duction to the quantum formalism, including the theory of quantum instruments and conditional probabilities.

2. Classical versus quantum probability

CP was mathematically formalized by Kolmogorov (1933). This is the calculus of probability measures, where a non-negative weight p(A) is assigned to any event A. The main property of CP is its additivity: if two events O 1 , O 2 are disjoint, then the probability of disjunction of these events equals to the sum of probabilities:

P (O 1 ∨ O 2 ) = P (O 1 ) + P (O 2 ).

QP is the calculus of complex amplitudes or in the abstract for- malism complex vectors. Thus, instead of operations on probability measures one operates with vectors. We can say that QP is a vector model of probabilistic reasoning. Each complex amplitude ψ gives the probability by the Born’s rule: Probability is obtained as the square of the absolute value of the complex amplitude.

p = |ψ| 2

(for the Hilbert space formalization, see Section 3.2, formula (7)). By operating with complex probability amplitudes, instead of the direct operation with probabilities, one can violate the basic laws of CP. In CP, the formula of total probability (FTP) is derived by using addi- tivity of probability and the Bayes formula, the definition of conditional P (O 2 ∩O 1 ) , P (O 1 ) > 0. Consider the pair, A and probability, P (O 2 |O 1 ) = P (O 1 ) B, of discrete classical random variables. Then ∑ P (B = β) = P (A = α)P (B = β|A = α).

Thus, in CP the B-probability distribution can be calculated from the A-probability and the conditional probabilities P (B = β|A = α). In QP, classical FTP is perturbed by the interference term (Khrennikov, 2010); for dichotomous quantum observables A and B of the von Neumann-type, i.e., given by Hermitian operators A ̂ and B, quantum version of FTP has the form:

P (B = β) = P (A = α)P (B = β|a = α) (1)

+2 ∑ cos θ α 1 α 2 √ P (A = α 1 )P (B = β|A = α 1 )P (A = α 2 )P (B = β|a = α 2 ) α 1 <α 2 P (O 1 ∨ O 2 ) = P (O 1 ) + P (O 2 ). (2)

If the interference term is positive, then the QP-calculus would gen- erate a probability that is larger than its CP-counterpart given by the classical FTP (2). In particular, this probability amplification is the basis of the quantum computing supremacy. There is a plenty of statistical data from cognitive psychology, deci- sion making, molecular biology, genetics and epigenetics demonstrat- ing that biosystems, from proteins and cells (Asano et al., 2015b) to hu- mans (Khrennikov, 2010; Busemeyer and Bruza, 2012) use this amplifi- cation and operate with non-CP updates. We continue our presentation with such examples.

3. Quantum instruments

3.1. A few words about the quantum formalism

Denote by  a complex Hilbert space. For simplicity, we assume that it is finite dimensional. Pure states of a system S are given by normalized vectors of  and mixed states by density operators (positive semi-definite operators with unit trace). The space of density operators is denoted by S(). The space of all linear operators in  is denoted by the symbol (). In turn, this is a linear space. Moreover, () is the complex Hilbert space with the scalar product, ⟨A|B⟩ = TrA ∗ B. We consider linear operators acting in (). They are called superoperators.

The dynamics of the pure state of an isolated quantum system is described by the Schrödinger equation: d ̂ ψ(t) = Hψ(t)(t), ψ(0) = ψ 0 , (3) dt where H ̂ is system’s Hamiltonian. This equation implies that the pure ̂ state ψ(t) evolves unitarily: ψ(t) = U ̂ (t)ψ 0 , where U ̂ (t) = e −it H is ̂ one parametric group of unitary operators, U (t) ∶  → . In quan- tum physics, Hamiltonian H ̂ is associated with the energy-observable. i The same interpretation is used in quantum biophysics (Arndt et al., 2009). However, in our quantum-like modeling describing information processing in biosystems, the operator H ̂ has no direct coupling with physical energy. This is the evolution-generator describing information interactions. Schrödinger’s dynamics for a pure state implies that the dynamics of a mixed state (represented by a density operator) is described by the von Neumann equation:

3.2. Von Neumann formalism for quantum observables

In the original quantum formalism (Von Neumann, 1955), physical ̂ We consider observable A is represented by a Hermitian operator A. ∑ only operators with discrete spectra: A ̂ = x x E ̂ A (x), where E ̂ A (x) is the projector onto the subspace of H corresponding to the eigenvalue x. Suppose that system’s state is mathematically represented by a density operator ρ. Then the probability to get the answer x is given by the Born rule Pr{A = x ∥ ρ} = Tr[ E ̂ A (x)ρ] = Tr[ E ̂ A (x)ρ E ̂ A (x)] and according to the projection postulate the post-measurement state is obtained via the state-transformation: E ̂ A (x)ρ E ̂ A (x) . (6) ρ → ρ x = T r E ̂ A (x)ρ E ̂ A (x)

For reader’s convenience, we present these formulas for a pure initial state ψ ∈ . The Born’s rule has the form:

The state transformation is given by the projection postulate:

Here the observable-operator A ̂ (its spectral decomposition) uniquely determines the feedback state transformations  A (x) for outcomes x

The map x →  A (x) given by (9) is the simplest (but very important) example of quantum instrument.

3.3. Non-projective state update: atomic instruments

In general, the statistical properties of any measurement are char- acterized by (:i) the output probability distribution Pr{x = x ∥ ρ}, the probability distribution of the output x of the measurement in the input state ρ;

(ii) the quantum state reduction ρ ↦ ρ {x=x} , the state change from

the input state ρ to the output state ρ {x=x} conditional upon the outcome x = x of the measurement.

In von Neumann’s formulation, the statistical properties of any mea- surement of an observable A is uniquely determined by Born’s rule (5) and the projection postulate (6), and they are represented by the map (9), an instrument of von Neumann type. However, von Neu- mann’s formulation does not reflect the fact that the same observable A represented by the Hermitian operator A ̂ in  can be measured in many ways. Formally, such measurement-schemes are represented by quantum instruments.

Now, we consider the simplest quantum instruments of non von Neumann type, known as atomic instruments. We start with recollection of the notion of POVM (probability operator valued measure); we restrict considerations to POVMs with a discrete domain of definition ̂ X = {x 1 , ... , x N , ...}. POVM is a map x → D(x) such that for each ̂ x ∈ X, D(x) is a positive contractive Hermitian operator (called effect) ̂ ∗ = D(x), ̂ ̂ (i.e., D(x) 0 ≤ ⟨ψ| D(x)ψ⟩ ≤ 1 for any ψ ∈ ), and the normalization condition ∑ ̂ D(x) = I

holds, where I is the unit operator. It is assumed that for any mea- surement, the output probability distribution Pr{x = x ∥ ρ} is given by ̂ Pr{x = x ∥ ρ} = Tr[ D(x)ρ],

where { D(x)} is a POVM. For atomic instruments, it is assumed that effects are represented concretely in the form

where V (x) is a linear operator in H. Hence, the normalization con- ∑ dition has the form x V (x) ∗ V (x) = I. The Born rule can be written similarly to (5):

Pr{x = x ∥ ρ} = Tr[V (x)ρV ∗ (x)]

It is assumed that the post-measurement state transformation is based on the map:

ρ →  A (x)ρ = V (x)ρV ∗ (x),

so the quantum state reduction is given by

The map x →  A (x) given by (13) is an atomic quantum instrument. We remark that the Born rule (12) can be written in the form Pr{x = x ∥ ρ} = Tr [ A (x)ρ].

Let A ̂ be a Hermitian operator in . Consider a POVM D ̂ = ( D ̂ A (x)) ̂ This POVM with the domain of definition given by the spectrum of A. represents a measurement of observable A if Born’s rule holds:

Thus, in principle, probabilities of outcomes are still encoded in the spectral decomposition of operator A ̂ or in other words operators D ̂ A (x) should be selected in such a way that they generate the probabilities corresponding to the spectral decomposition of the symbolic represen- tation A ̂ of observables A, i.e., D ̂ A (x) is uniquely determined by A ̂ as D ̂ A (x) = E ̂ A (x). We can say that this operator carries only information about the probabilities of outcomes, in contrast to the von Neumann scheme, operator A ̂ does not encode the rule of the state update. For an atomic instrument, measurements of the observable A has the unique output probability distribution by the Born’s rule (16), but has many different quantum state reductions depending of the decomposition of ̂ the effect D(x) = E ̂ A (x) = V (x) ∗ V (x) in such a way that

Bibliography & references