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{{FR
| Title = Quantum-like modeling in biology with open quantum systems and instruments
| author1 = Irina Basieva
| author2 = Andrei Khrennikov
| author3 = Masanao Ozawa
| Source = http://crossmark.crossref.org/dialog/?doi=10.1016/j.biosystems.2020.104328&domain=pdf<!-- where this work comes from or where was it was retrieved (URL) -->
| Original = https://www.sciencedirect.com/science/article/pii/S0303264720301994?via%3Dihub<!-- link to the original screenshot or PDF print from the retrieval -->
| Date = 27 October 2020<!-- date of the original work, when the author/s published it (dd/mm/yyyy) -->
| Journal = BioSystems
| DOI = 10.1016/j.biosystems.2020.104328
| ISBN =
| PUBMED =
| Pdfcopy = <!-- eventual polished PDF document -->
| License = CC BY
}}

== 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.<ref>We recall that Wigner emphasized an enormous effectiveness of mathematics in physics (Wigner, 1960). But, famous Soviet mathematician Gelfand contrasted
the Wigner’s thesis by pointing to ‘‘surprising ineffectiveness of mathematics in biology’’ (this remark was mentioned by another famous Soviet mathematician
Arnold (Arnol’d, 1998) with reference to Gelfand).</ref><ref>In particular, this special issue contains the article on the use of p-adic numbers and analysis in mathematical biology (Dragovich et al., 0000). This is
non-straightforward generalization of the standard approach based on the use of real numbers and analysis.</ref>

In this paper, we present the applications of the mathematical
formalism of quantum mechanics and its methodology to modeling
biosystems’ behavior.<ref>We are mainly interested in quantum measurement theory (initiated in Von Neumann’s book (Von Neumann, 1955)) in relation with theory of quantum
instruments (Davies and Lewis, 1970; Davies, 1976; Ozawa, 1984; Yuen, 1987; Ozawa, 1997, 2004; Okamura and Ozawa, 2016) and theory of open quantum
systems (Ingarden et al., 1997).</ref> 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.<ref>See Khrennikov (1999, 2003) and Khrennikov (2004b) for a few pioneer
papers, Khrennikov (2004a, 2010), Busemeyer and Bruza (2012), Asano et al.
(2015b) and Bagarello (2019) for monographs, and Haven (2005), Khrennikov
(2006), Busemeyer et al. (2006), Pothos and Busemeyer (2009), Dzhafarov
and Kujala (2012), Bagarello and Oliveri (2013), Wang and Busemeyer
(2013), Wang et al. (2014), Khrennikov and Basieva (2014b), White et al.
(2014), Khrennikov and Basieva (2014a), Khrennikova (2014), Busemeyer
et al. (2014), Boyer-Kassem et al. (2015), Dzhafarov et al. (2015), Khrennikov
(2016b), Khrennikova (2016), Asano et al. (2017b,a), Khrennikova (2017),
Surov et al. (2019) and Basieva et al. (2017) for some representative papers.</ref> 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).<ref>CP-calculus is closed calculus of probability measures. In QP-
calculus (Von Neumann, 1955; Khrennikov, 2016a), probabilities are
not the primary objects. They are generated from quantum states with the
aid of the Born’s rule. The basic operations of QP-calculus are presented in
terms of states, not probabilities. For example, the probability update cannot
be performed straightforwardly and solely with probabilities, as done in CP
— with the aid of the Bayes formula.</ref>
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).<ref>The Kolmogorov probability space (Kolmogorov, 1933) is any triple
(Ω,  , P), where Ω is a set of any origin and  is a σ-algebra of its subsets, P
is a probability measure on  . The set Ω represents random parameters of the
model. Kolmogorov called elements of Ω elementary events. Sets of elementary
events belonging to  are regarded as events.</ref> 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<ref>We recall that interference is the basic feature of waves, so often one
speaks about probability waves. But, one has to be careful with using the
wave-terminology and not assign the direct physical (or biological) meaning
to probability waves.</ref> 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.<ref>Say A ̂ = H ̂ is the operator representing the energy-observable. This is just
a theoretical entity encoding energy. Energy can be measured in many ways
within very different measurement schemes.</ref> 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.<ref>We remark that any projector E ̂ is Hermitian and idempotent, i.e., E ̂ ∗ = E ̂
̂ Thus, any projector E ̂ A (x) can be written as (11): E ̂ A (x) =
and E ̂ 2 = E.
E ̂ A (x) ∗ E ̂ A (x). The map x → E ̂ A (x) is a special sort of POVM, the projector
valued measure — PVM, the quantum instrument of the von Neumann type.</ref> 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
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