# Quantum-like model for unconscious-conscious interaction and emotional
coloring of perceptions and other conscious experiences

Andrei Khrennikov
Linnaeus University, International Center for Mathematical Modeling
in Physics and Cognitive Sciences Växjö, SE-351 95, Sweden

###### Abstract

Quantum measurement theory is applied to quantum-like modeling of coherent
generation of perceptions and emotions and generally for emotional coloring of
conscious experiences. In quantum theory, a system should be separated from an
observer. The brain performs self-measurements. To model them, we split the
brain into two subsystems, unconsciousness and consciousness. They correspond
to a system and an observer. The states of perceptions and emotions are
described through the tensor product decomposition of the unconscious state
space; similarly, there are two classes of observables, for conscious
experiencing of perceptions and emotions, respectively. Emotional coloring is
coupled to quantum contextuality: emotional observables determine contexts.
Such contextualization reduces degeneration of unconscious states. The
quantum-like approach should be distinguished from consideration of the
genuine quantum physical processes in the brain (cf. Penrose and Hameroff). In
our approach the brain is a macroscopic system which information processing
can be described by the formalism of quantum theory.

,

keywords: Quantum-like model, unconscious-conscious interaction, emotional
coloring, perceptions, quantum measurement theory

##  1 Introduction

We start the paper with citation from [1]: “Although emotions, or feelings,
are the most significant events in our lives, there has been relatively little
contact between theories of emotion and emerging theories of consciousness in
cognitive science.” We want to formalize this contact in the quantum-like
framework for cognition by generalizing the sensation-perception model based
on quantum information theory [2].

In the present paper, the quantum theory of measurement is applied to quantum-
like modeling of coherent generation of perceptions and emotions and generally
for emotional coloring of conscious experiences. In quantum theory, a system
should be separated from an observer (or a measurement apparatus) [3]. In the
quantum framework, this separation is very important – although establishing
the sharp boundary between a system and a measuring apparatus is a difficult
problem.111It becomes even more difficult if a conscious observer,
experimenter, is involved into consideration of a measurement process. The
latter was done by Wigner. He considered a physical system interacting with a
measurement apparatus. The latter is under monitoring of an observer-
experimenter whose nervous system detects and propagates the signal, e.g.,
visual, obtained from apparatus’ pointer. By Wigner [4, 5] (see also [6]), a
measurement is completed by consciousness So, for him consciousness played the
crucial role in the measurement process. The majority of the quantum community
did not share Wigner’s viewpoint; for them physical measurement is terminated
at the pointer of an apparatus. The brain, as a physical system, performs so
to say self-measurements. To model such self-measurement, we split the brain,
as an information processor, into two subsystems, unconsciousness ${\cal UC}$
and consciousness ${\cal C}.$ The former plays the role of a system under
observation and the latter of an observer [2] (see [7]-[10] for mathematical
modeling of join functioning of unconsciousness and unconsciousness based on
treelike geometry of the brain). To model the cooperation of perceptions and
emotions, the state space of ${\cal UC}$ is decomposed into the tensor product
of corresponding state spaces,

${\cal H}_{{\cal UC}}={\cal H}_{\rm{per}}\otimes{\cal H}_{\rm{em}}.$

Similarly, there are considered two classes of observables ${\cal A}$ and
$O_{\rm{sup}},$ for conscious experiencing of perceptions and emotions,
respectively.

This paper is concentrated on modeling of emotional coloring of perceptions.
But, we present the very general scheme of coloring of one class of conscious
experiences with another, “basic experiences” are colored with “supplementary
experiences”. The aim and the origin of such coloring will be discussed below.

Emotional coloring is modeled in the framework of quantum contextuality \-
emotional observables determine contexts for perceptions and other basic
conscious experiences. We highlight that contextualization is a way to reduce
degeneration of unconscious states. Such contextual reduction is very
important, since the state space of unconsciousness ${\cal H}_{{\cal UC}}$ has
huge dimension and each conscious experience $x$ is based on multidimensional
subspace ${\cal H}_{x}$ of ${\cal H}_{{\cal UC}}.$

We stress that quantum-like modeling of brain’s functioning should be sharply
distinguished from theories based on consideration of genuine quantum physical
processes in the brain (cf. [11] – [20]). In the quantum-like approach, the
brain is a macroscopic system which information processing can be described by
the mathematical formalism of quantum theory (cf. Gunji et al. [21, 22]). The
quantum-like cognition project (see, e.g., monographs [9], [23] – [29]) does
not contradict to the quantum cognition project. However, we proceed without
the assumption that quantum features of information processing by the brain
are coupled to quantum physical processes. The main distinguishing feature of
the quantum information processing is operating with superpositions of
alternatives. Such operating can be physically realized with classical
electromagnetic waves. Genuine quantum information features are coupled to
measurements generating discrete events from superpositions. 222In Bohr’s
works [3], such an event was called phenomenon and its individuality was
emphasized (see [30] on comparison of quantum information processing in
classical vs. quantum optics).

We treat brain’s functioning in the purely informational framework, so the
states are not physical (electrochemical) states, but the information states.
We do not need to localize ${\cal UC}$ and ${\cal C}$ in the concrete brain’s
areas. Of course, neuro-physiological studies give us coupling of ${\cal UC}$
and ${\cal C}$ to some areas in the brain, e.g., ${\cal C}$ to the prefrontal
cortex. However, the distributed and physically nonlocalized information
processing matches better our quantum-like model. We neither try to describe
the neurophysiological mechanism of generation of perceptions and emotions and
“coloring” of the former with the latter (see, e.g., [31] – [33] for details).

Of course, information processing in the brain is indivisibly coupled to
electrochemical processes. However, as was emphasized by Liberman et al. [34]
– [37] and Igamberdiev and Brenner [38], physical reduction of consciousness
is impossible.333See, e.g., paper [34]: “Biophysics cannot use the ordinary
laws of physics and must take into account the influence on the phenomena to
be studied, not only of a measurement but also of a calculation process in the
real device predicting the future. See, also [37] “Living organisms measure
many parameters in order to have orientation in the outer medium. That is why
biophysics cannot use the ordinary laws of physics and must take into account
the influence on the phenomena to be studied not only of a measurement but
also of a calculation process in the real physical and biophysical device
predicting the future.”

##  2 The basic theories of consciousness

As is well known, there are two basic competing theories of consciousness:

  * •

the First Order Theory of Consciousness [41] – [45];

  * •

the Higher Order Theory of Consciousness [46] – [48].

We characterize these theories with the following citation from article [1]:

“First-order theorists, such as Block, argue that processing related to a
stimulus is all that is needed for there to be phenomenal consciousness of
that stimulus [41] – [45]. Conscious states, on these kinds of views, are
states that make us aware of the external environment. Additional processes,
such as attention, working memory, and metacognition, simply allow cognitive
access to and introspection about the first-order state. In the case of visual
stimuli, the first-order representation underlying phenomenal consciousness is
usually said to involve the visual cortex, especially the secondary rather
than primary visual cortex. Cortical circuits, especially involving the
prefrontal and parietal cortex, simply make possible cognitive (introspective)
access to the phenomenal experience occurring in the visual cortex.”

“In contrast, David Rosenthal and other higher-order theorists argue that a
first-order state resulting from stimulus-processing alone is not enough to
make possible the conscious experience of a stimulus. In addition to having a
representation of the external stimulus one also must be aware of this
stimulus representation. This is made possible by a HOR, which makes the
first-order state conscious. In other words, consciousness exists by virtue of
the relation between the first- and higher-order states. Cognitive processes,
such as attention, working memory, and metacognition are key to the conscious
experience of the first-order state. In neural terms, the areas of the GNC,
such as the prefrontal and parietal cortex, make conscious the sensory
information represented in the secondary visual cortex.”

The Higher Order Theory distinguishes between unconscious and conscious
processing of mental information in the brain. By this theory, what makes
cognition conscious is a higher-order observation of the first-order
processing. And in quantum theory observation is not simply inspection of the
system’s state. This is a complex process of interaction between a system and
a measurement device. This is the good place to cite Bohr [3]:

“This crucial point … implies the impossibility of any sharp separation
between the behaviour of atomic objects and the interaction with the measuring
instruments which serve to define the conditions under which the phenomena
appear. In fact, the individuality of the typical quantum effects finds its
proper expression in the circumstance that any attempt of subdividing the
phenomena will demand a change in the experimental arrangement introducing new
possibilities of interaction between objects and measuring instruments which
in principle cannot be controlled.”

This viewpoint matches better with the Higher Order Theory of Consciousness. A
conscious experience is not simply introspection of the ${\cal UC}$-state.

In this paper, we do not describe the process of ${\cal UC}-{\cal C}$
interaction. In particular, we do not operate with the states of ${\cal C}.$
This can be done similarly to paper [2] by using the scheme of indirect
quantum measurements in the framework of open quantum systems theory. However,
it would make the presentation essentially more complicated from the
mathematical viewpoint and the basic issue of this paper, contextuality, would
be overshadowed by technicalities. We apply straightforwardly the canonical
quantum observation theory as it was made by von Neumann [49].444In [20]
Igamberdiev considered another approach to quantum measurement theory, namely
non-demolition quantum measurements, as the basis of bio-information
processing. Such measurements differ from the von Neumann measurements. In
some sense, they are closer to classical measurements of electromagnetic
waves. Although he referred to genuine physical processes, he emphasized that
in biosystems these processes are lifted to the macrolevel: “The smallest
details of living systems are molecular devices that realize non-demolition
quantum measurements. These smaller devices form larger devices
(macromolecular complexes), up to living body. The quantum device possesses
its own potential internal quantum state (IQS), which is maintained for a
prolonged time via reflective error-correction.

The latter is slightly generalized by representation of observables by
projector valued measures, instead of Hermitian operators. This generalization
is motivated by two things. Starting with a Hermitian operator, it is possible
to find its spectral decomposition and the projectors onto the subspaces
corresponding to the eigenvalues, but the subspaces appear as secondary
objects. In the POV representation of observers, subspaces are fundamental,
and observer’s values are associated with them. In our model, conscious
experiences are associated with concrete subspaces of ${\cal H}_{{\cal UC}}.$
Another advantage of using POV-observables is that we are not obliged to label
the values of observables by real numbers (operator’s eigenvalues); any
vocabulary can be used for description of conscious experiences.

We remark that, although Bohr’s viewpoint on the outcomes of quantum
measurements dominates, a few respectable scientists claimed that these
outcomes can be considered as the objective properties of physical systems.
Quantum measurements are treated as just approaching the premeasurement values
of observables. This position was presented in the well know paper Einstein,
Podolsky, and Rosen (EPR) [50]; later Bell elaborated (and modified) EPR’s
argument [51, 52], but he confronted the problem of nonlocality. This line of
thought matches better to the First Order Theory of Consciousness. We shall
follow the Bohr’s line of thought [53] and hence couple quantum measurement
theory with the Higher Order Theory of Consciousness.

In application to emotions, the First Order Theory of Consciousness matches
the somatic theories of emotions rooted to James [54]; the first of them is
James-Lange theory [55]. Nowadays this viewpoint on emotions is advertized by
some prominent neuro-physiologists, e.g. Damasio [56].

##  3 Perceptions and emotions

###  3.1 Perception representation of sensations

We follow to von Helmholtz [57] theory of sensation-perception. Perceptions
are not simply a copies of sensations, not “impressions like the imprint of a
key on wax”, but the results of complex signal processing including
unconscious cognitive processing. In the modern science formulation, the
process of creation of perception can be described as follows: “Sensory
information undergoes extensive associative elaboration and attentional
modulation as it becomes incorporated into the texture of cognition. This
process occurs along a core synaptic hierarchy which includes the primary
sensory, upstream unimodal, downstream unimodal, heteromodal, paralimbic and
limbic zones of the cerebral cortex.” [58]

###  3.2 Context-representation via emotions

As is emphasized in [1], “Emotion schema are learned in childhood and used to
categorize situations as one goes through life. As one becomes more
emotionally experienced, the states become more differentiated: fright comes
to be distinguished from startle, panic, dread, and anxiety.” In our
terminology, each emotion-generation scheme is crystallized on of the basic
life-contexts. Context-labeling is the basic function of emotions.
Contextualization of surrounding environment was one of the first cognitive
tasks of biosystems and this ability was developed in parallel with
establishing of sensation-perception system.

Memory is heavily involved in emotional activity, both for memorizing the
features of contexts and for comparison of new perceptions with these
contexts. (See, e.g., [31] for the memory’s role in cognition). Thus,
evolutionary there was designed a mental information processing system
representing the basic life-contexts. This system is fixed at the level of the
brain (and more generally the nervous system) hardware. But, memorizing a
variety of contextual experiences is done on the basis of the experiencing
various situations (see above citation from [1]). This context-refection
system was the root of the present emotion-system in humans. The latter has
complex cognitive functions, not only contextual. However, in this paper we
concern mainly contextuality.

Emotions represent adaptive reactions to environmental challenges; they are a
result of human evolution; they provided optimal (from the viewpoint of
computational resources) solutions to ancient and recurring problems that
faced our ancestors [59].555Although we do not follow the James-Lange theory
of emotions [55], this is the good place to mention that James [54] pointed
out that “feeling of the same changes as they occur is the emotion. Here, for
us the key words are “the same changes as they occur”, i.e., the complex of
repeatable bodily changes - physiological encoding of a context.

We emphasize that in our model emotions are conscious, cf. [1]: “One
implication of our view is that emotions can never be unconscious. Responses
controlled by subcortical survival circuits that operate nonconsciously
sometimes occur in conjunction with emotional feelings but are not emotions.
An emotion is the conscious experience that occurs when you are aware that you
are in particular kind of situation that you have come, through your
experiences, to think of as a fearful situation. If you are not aware that you
are afraid, you are not afraid; if you are not afraid, you aren’t feeling
fear.”

##  4 Unconscious vs. conscious information processing in the brain

###  4.1 Unconsciousness

An essential part of information processing in the brain is performed
unconsciously; the information system responsible for such processing (call it
unconsciousness) is denoted by the symbol ${\cal UC}.$ The space of its states
is denoted by ${\cal H}\equiv{\cal H}_{{\cal UC}}.$ In the quantum-like model,
this is a complex Hilbert space (section 5).

The reader need not couple the notion of unconsciousness with the names of
James [60], Freud [61], and Jung [62] (although the author of this paper was
strongly influenced by them, cf. with the previous works [7] – [10]). In this
paper, ${\cal UC}$ denotes a special information processors of the brain. It
performs pre-observational processing of the mental state.

###  4.2 Consciousness

Perceptions and emotions are commonly treated as conscious entities. So, in
our model the brain contains another information processing system generating
conscious experiences; denote it by the symbol ${\cal C}.$ In our quantum-like
framework, its functioning is modeled as performing measurements on the system
${\cal UC}.$ Introduction of two systems ${\cal UC}$ and ${\cal C}$ matches
the quantum measurement scheme, ${\cal UC}$ is the analog of a physical system
exposed to measurements and ${\cal C}$ is the analog of a complex of
measurement apparatuses.

In the operational quantum approach, measurable quantities, observables, are
represented by Hermitian operators acting in system’s state space. More
advanced models within theory of open quantum systems are based on
consideration of the states of a measurement apparatus and interaction between
the states of the system and apparatus [63] – [65]. In our quantum-like
modeling, the latter corresponds to consideration of the conscious states
represented in complex Hilbert space ${\cal H}_{{\cal C}}.$ In this paper we
shall not consider (cf. [2, 66] for modeling the process of ${\cal UC}-{\cal
C}$ interactions and generation of outputs of conscious observables).

Unconscious-conscious modeling of the brain’s functioning matches well to the
philosophic paradigm of the ontic-epistemic structuring of scientific
theories. The ontic level is about reality (physical and mental) as it is if
nobody observes it, the epistemic level describes observations (see
Atmanspacher [67]). In contrast to quantum physics, in our model the epistemic
level is related not to external observers, but to brain’s self-observations.
In the brain, unconscious and conscious processes are closely coupled and the
sharp separation between them is impossible (cf. Brenner [68] and Brenner and
Igamberdiev [69].

###  4.3 Unconscious and conscious counterparts of the processes of generation
of perceptions and emotions

In this paper, we shall be mainly concentrated on functioning of two
information processors transforming

  * •

sensations $\to$ perceptions,

  * •

contexts $\to$ emotions.

Both processors have conscious outputs. Their functioning is strongly
correlated; in the formalism quantum theory correlations are represented by
entangled states. We denote unconscious counterparts of these processors by
the symbols ${\cal UC}_{\rm{per}}$ and ${\cal UC}_{\rm{em}},$ respectively. In
modeling of the emotional coloring of perception (its contextualization), we
shall consider the compound information system $({\cal UC}_{\rm{per}},{\cal
UC}_{\rm{em}}).$

This is the good place to mention the first theory of emotions, the James-
Lange theory [55]. James claimed that “I am trembling. Therefore I am afraid.”
He stated: “My thesis … is that the bodily changes follow directly the
perception of the exciting fact and that our feeling of the same changes as
they occur is the emotion.” This way of thinking matches with the First Order
Theory of Consciousness and the EPR-Bell viewpoint on quantum measurements.

Following LeDoux [31] (see also [1]), we treat emotions within the Higher
Order Theory of Consciousness coupled to Bohr’s interpretation of quantum
measurements.

###  4.4 Basic and supplementary conscious experiences, application to
decision making

Although we are mainly interested in emotional coloring of perceptions, the
formalism under consideration can be applied to the very general class of
compound information processing systems, $({\cal UC}_{\rm{bas}},{\cal
UC}_{\rm{sup}}).$ The latter is used for determining stable repeatable and
evolutionary fixed contexts for the former. Simplest generalization of the
perception-emotion scheme is emotional contextualization of decision making
which modeling is based the compound system $({\cal UC}_{\rm{dm}},{\cal
UC}_{\rm{em}}).$

Generation of conscious experiences (basic and supplementary) is modeled with
POV-observables; denote the corresponding classes by the symbols
$O_{\rm{bas}}$ and $O_{\rm{sup}}.$ In particular, we shall consider the pairs
$(O_{\rm{per}},O_{\rm{em}})$ and $(O_{\rm{dm}},O_{\rm{em}}).$

An expert in cognition may suggest other pairs of basic and supplementary
conscious experiences. One of the properties of the supplementary mental
experiences is their rapid processing. They should not inhibit processing of
the basic mental experiences. We remark that emotion’s generation is
characterized by high speed (see [31]).

##  5 Quantum formalism: states and observables

Denote by ${\cal H}$ 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 ${\cal H}.$ Later we shall consider mixed states (section ), but,
for the moment, we proceed with only pure states and call them simply statets.

Physical observable $A$ is represented by a Hermitian operator denoted by the
same symbol. Consider an operator with discrete spectrum; its spectral
decomposition has the form:

$\hat{A}=\sum_{x}x\;E^{A}_{x},$ (1)

where $E^{A}_{x}$ is the orthogonal projector onto the subspace of $H$
corresponding to the eigenvalue $x,$ i.e., ${\cal H}^{A}_{x}=E^{A}_{x}{\cal
H}.$ We recall that the spectral family of orthogonal projectors satisfies the
normalization condition

$\sum_{x}x\;E^{A}_{x}=I,$ (2)

where $I$ is the unit operator, and the mutual orthogonality condition

$E^{A}_{x}\perp E^{A}_{y},x\not=y,$ (3)

or equivalently

$E^{A}_{x}E^{A}_{y}=\delta(x-y)E^{A}_{x}.$ (4)

The probability to get the answer $x$ for a pure initial state $\psi$ is given
by the Born rule

$\Pr\\{A=x\|\psi\\}=\|E^{A}_{x}\psi\|^{2}=\langle\psi|E^{A}_{x}\psi\rangle.$
(5)

and according to the projection postulate (von Neumann [49]) the post-
measurement state is generated by the map:

$\psi\to{\cal I}^{A}_{x}\psi=E^{A}_{x}\psi/\|E^{A}_{x}\psi\|.$ (6)

This state transformation is generated by observation’s feedback to the system
which initially was in the state $\psi,$ i.e., observations disturb systems’
states.

The quantum state update is the basis of quantum generalization of classical
Bayesian inference . The projection update has properties which crucially
differ from the classical probability update. In particular, it generates
violation of the law of total probability (playing the important role in
Bayesian inference) [70, 9, 71] and the order effect which is absent in the
classical probability theory (see Wang and Busemeyer [72], Ozawa and
Khrennikov [66] for its quantum-like modeling). The author of the present
paper have stressed many times that non-Bayesian character of the quantum
state and probability update is one of the distinguishing features of the
quantum-like modeling of the brain’s functioning [73] (cf. Gunji et al. [21,
22]).

In physics, the values of observables are labeled by real numbers and it is
convenient to represent observables by linear operators. But, in fact, the
basic formulas (5), (6) contain only the orthogonal projectors $(E^{A}_{x})$
and association of them with real numbers is not crucial. The label $x$ can
belong to any set $X$ and then the probability distribution given by Born’s
rule (5) is defined on the set $X.$ The family of orthogonal projectors
$(E^{A}_{x})_{x\in X}$ encoding an observable $A$ should satisfy two
constraints (2), (4). They guarantee that, for any state $\psi,$ quantity
$\Pr\\{A=x\|\psi\\}$ determined by Born’s rule (5) is a probability measure.
(For simplicity, we consider only discrete sets of labels.) Such family of
projectors $(E^{A}_{x})_{x\in X}$ is the simplest example of a positive
operator valued measure (POVM), namely, the projector-valued measure (POV).
The state transformation map (6), $\psi\to{\cal I}^{A}_{x}\psi,$ determines
the simplest quantum instrument [63]– [65]. The symbol $A$ just labels an
observable (not Hermitian operator), $A=(E^{A}_{x}).$ In principle, the same
scheme can be realized with general quantum instruments and POVMs (see [63] –
[65]), but for simplicity we proceed with POVs.

##  6 Incompatible conscious observables

In quantum physics, observables $C_{1}$ and $C_{2}$ are called compatible if
they can be jointly measurable and the joint probability distribution (JPD)
$p_{\psi}(C_{1}=x_{1},C_{2}=x_{2})$ is well defined; observables which cannot
be jointly measurable and, hence, their JPD cannot be defined are called
incompatible. In the mathematical formalism, compatibility and incompatibility
are formalized through commutativity and noncommutativity, respectively. If
observables are described as Hermitian operators $C_{1},C_{2},$ compatibility
is encoded as $[C_{1},C_{2}]=0;$ if they are POV-observables, then
compatibility is encoded as

$[E^{C_{1}}_{x_{1}},E^{C_{2}}_{x_{2}}]=0,\;\mbox{for all}\;x_{1},x_{2}.$ (7)

Incompatibility is encoded as $[C_{1},C_{2}]\not=0$ or, for POV-observables,
as violation of (7) at least for one pair $(x,y).$

For compatible observables, JPD is given by the following extension of the
Born’s rule:

$p_{\psi}(C_{1}=x_{1},C_{2}=x_{2})=|\langle
E^{C_{1}}_{x_{1}}E^{C_{2}}_{x_{2}}\psi,\psi\rangle|^{2}=|\langle
E^{C_{2}}_{x_{2}}E^{C_{1}}_{x_{1}}\psi,\psi\rangle|^{2}.$ (8)

By using JPD compatible observables can be modeled in the classical
probabilistic formalism. The formula (9) can be generalized for an arbitrary
number of observables $C_{1},...,c_{m},$ as

$p_{\psi}(C_{1}=x_{1},...,C_{m}=x_{m})=|\langle
E^{C_{1}}_{x_{1}}....E^{C_{2}}_{x_{m}}\psi,\psi\rangle|^{2}$ (9)

and this expression is invariant w.r.t. permutations.

We stress that the space of observables $O_{\rm{per}}$ can contain
incompatible perceptions as well as $O_{\rm{em}}$ can contain incompatible
emotions. In physics incompatibility is often seen as the exotic property of
quantum theory - comparing with classical physical theory. Philosophically it
is represented in the Bohr’s complementarity principle which is difficult for
understanding and was many times reformulated by Bohr [3] . However, in the
mental framework the notion of incompatibility can be interpreted very
naturally: there exist say emotions which can be experienced simultaneously;
say happiness and sadness, pride and shame; in the same way it is evident that
there exist incompatible, i.e., jointly unobservable perceptions and other
conscious experiences. We note that mental observations are brain’s self-
observations. May be this self-observational property simplifies the
incompatibility issue.

The necessity to operate with various incompatible entities is the main roots
of the use of the quantum(-like) information representation. In the absence of
incompatibility, i.e., if, for the same mental state, the brain were able to
construct the consistent probabilistic representation (in the form of JPD) of
all possible combinations of say emotions, the quantum state formalism would
be unnecessary.

##  7 Resolution of degeneration of states and quantum contextuality

Consider an observable $A$ which is mathematically described as POV
$(E^{A}_{x})_{x\in X}.$ For the discrete set of observance’s values
$X=(x_{k}),$ we use notation $E_{k}^{A}=E_{x_{k}}^{A}.$ Suppose now that some
of its projections $E_{k}^{A}$ are degenerate; $\rm{dim}\;{\cal H}_{k}^{A}>1,$
where ${\cal H}_{k}^{A}=E_{k}^{A}{\cal H}.$ Moreover, for some outcomes,
degeneration is very high, $\rm{dim}\;{\cal H}_{k}^{A}>>1.$ In this case, a
huge set of states corresponds to the same outcome $x_{k}.$

Observer may be unsatisfied by such a situation; observer wants to refine his
observation to split (at least partially) the states corresponding to the
fixed outcome. How can it be done? The quantum measurement formalism presents
the very natural and simple procedure for refinement of states’ structure.

Consider another quantum POV-observable $B=(E_{m}^{B})$ compatible with the
original observable $A,$ i.e., $[E_{k}^{A},E_{m}^{B}]=0$ for all indexes
$k,m.$ We remark that $B$-observable has its own set of outcomes, $Y=(y_{m}),$
which need not coincide with $X.$

For any (pure) quantum state $\psi,$ the POV-observables $A$ and $B$ can be
jointly measurable with outcomes $(x_{k},y_{m})$ and the join probability
distribution

$p_{AB}(x_{k},y_{m}|\psi)=\|E_{k}^{A}E_{m}^{B}\psi\|^{2}=\|E_{m}^{B}E_{k}^{A}\psi\|^{2}.$
(10)

The projection postulate and commutativity of projectors imply that the post-
measurement state is the same for the joint measurement with outcome
$(x_{k},y_{m})$ and sequential measurements, first $A=x_{k}$ and then
$B=y_{m}$ or vice verse:

$\psi\to{\cal I}^{A}_{x}{\cal
I}^{B}_{y}\psi=\frac{E^{A}_{x}E^{B}_{y}\psi}{\|E^{A}_{x}E^{B}_{y}\psi\|}=\frac{E^{B}_{y}E^{A}_{x}\psi}{\|E^{B}_{y}E^{A}_{x}\psi\|}={\cal
I}^{B}_{y}{\cal I}^{A}_{x}\psi.$ (11)

The state space ${\cal H}_{k}^{A}$ is reduced to the space ${\cal
H}_{km}^{(A,B)}=E_{k}^{A}E_{m}^{B}{\cal H}(=E_{m}^{B}E_{k}^{A}{\cal H});$ so

${\cal H}_{k}^{A}=\oplus_{m}{\cal H}_{km}^{(A,B)}$ (12)

(direct sum decomposition). Thus, the observer can specify its post-
observation state much better.

The most fruitful refinement is based on an observable with nondegenerate
spectra. Let all projectors ${\cal I}^{B}_{m}$ be one dimensional,
$\rm{dim}\;{\cal H}_{m}^{B}=1$ and let $e_{m}^{B}$ be the corresponding basis
vector, i.e., $E_{j}^{B}e_{m}^{B}=\delta(j-m)e_{m}^{B}.$ In this case, the
outcome $(x_{k},y_{m})$ completely determines the post-observation mental
state: $\psi\to e_{m}^{B}.$ The correspondence between labels $(x_{k},y_{m})$
and post-measurement states is one-to-one.

Consider now another observable $C$ which is also compatible with $A$,
mathematically this is expressed as $[E^{A}_{k},E^{C}_{n}]=0$ for all $k,n.$
The state spaces corresponding to $A$-outcomes can also be refined w.r.t.
$C$-outcomes, i.e.,

${\cal H}_{k}^{A}=\oplus_{m}{\cal H}_{km}^{(A,C)},$ (13)

where ${\cal H}_{km}^{(A,C)}=E_{k}^{A}E_{m}^{C}{\cal
H}(=E_{m}^{C}E_{k}^{A}{\cal H}).$ In particular, if, for all $n,$
$\rm{dim}\;{\cal H}_{n}^{C}=1$ with the basis vector $e_{m}^{B},$ i.e.,
$E_{j}^{C}e_{n}^{C}=\delta(j-m)e_{n}^{C},$ then the outcome $(x_{k},z_{n})$
completely determines the post-observation mental state: $\psi\to e_{n}^{C}.$

In quantum measurement theory, selection of observables co-measurable with $A$
is considered as specification of measurement context of $A$-measurements; the
$A$-value in the $B$-context can differ from the $A$-value in the $C$-context,
for the same premeasurment state $\psi.$ This is the essence of contextuality
playing so important role in quantum information theory [74]:

Definition 1. If $A,B,C$ are three quantum observables, such that $A$ is
compatible with $B$ and $C,$ a measurement of $A$ might give different result
depending upon whether $A$ is measured with $B$ or with $C.$

We note that contextual behavior corresponds to the case of incompatible
quantum observables $B$ and $C$, i.e., there exist indexes such that
$[E^{B}_{m},E^{C}_{n}]\not=0.$ If all observables are pairwise commute, i.e.,
for all indexes
$[E^{A}_{k},E^{B}_{m}]=[E^{A}_{K},E^{C}_{n}]=[E^{B}_{m},E^{C}_{n}]=0,$ then,
for any state $\psi,$ it is possible to construct the noncontextual model of
measurement based on the joint probability distribution for triple outcomes

$p_{ABC}(x_{k},b_{m},c_{n}|\psi)=\|E_{k}^{A}E_{m}^{B}E_{n}^{C}\psi\|^{2}=...=\|E_{k}^{A}E_{n}^{C}E_{m}^{B}E_{n}^{C}\psi\|^{2}.$
(14)

If $B$ and $C$ are incompatible, such a model is impossible. This is the
contextuality scenario. However, contextuality formalized via Definition 1
cannot be tested experimentally, since it involves counterfactual reasoning.
The only possibility is test contextuality (based on Definition 1) indirectly
with the aid of Bell-type inequalities (see appendix).

If the sets of observables’ outcomes coincide with subsets of the real line,
then the above considerations can be essentially simplified - with the
Hermitian linear operators representing observables. The contextuality
scenario is related to observables satisfying the commutation relations
$[A,B]=[A,C]=0,[B,C]\not=0.$

However, the operator language is misleading and not only because
representation of outcomes by real numbers is too special for coming cognitive
applications. The main problem is that in the linear-operator approach the
basic entities are outcomes, eigenvalues of the operator-observable. The
subspace ${\cal H}^{A}_{x}$ is constructed as the space of the eigenvectors of
the operator $A$ corresponding to the eigenvalue $x.$ In our coming modeling,
we proceed another way around. The basic structures are mutually orthogonal
subspaces ${\cal H}_{k}$ such that

${\cal H}=\oplus_{k}{\cal H}_{k}.$ (15)

Then each of these subspaces is identified with some value of the POV-
observable $A$ given by projectors on these subspaces. Roughly speaking, first
states then values.

##  8 Resolution of degeneration of the states of consciousness via contextual
coloring

We start with the presentation of the general scheme for the resolution of
degeneration of the ${\cal C}$-states. In this scheme ${\cal C},$ operates
with two classes of observables $O_{\rm{bas}}$ and $O_{\rm{sup}}$ representing
the “basic and supplementary conscious experiences”, respectively.

In the quantum-like model of generation of conscious experiences developed by
the author [2], consciousness ${\cal C}$ is modeled as a system performing
observations over unconsciousness ${\cal UC}.$ As in section 4.1, the symbol
${\cal H}$ denotes the space of ${\cal UC}$-states. A conscious observable $A$
is represented as a POV-observable $E^{A}_{x},x\in X,$ where $X$ is the set of
conscious experiences. The latter can be, for example, the set of language
expression or visual images. The values of an observable are determined by the
subspaces ${\cal H}^{A}_{x}.$ If consciousness ${\cal C}$ detects a state
belonging to ${\cal H}^{A}_{x},$ then it generates the conscious experience
$x\in X.$

The space of unconscious states ${\cal H}$ has very high dimension. In
reality, it is infinite dimensional, since this is the quantum information
representation of electrochemical waves in the brain (see [75] for details).
We restrict modeling to the finite dimensional case for state spaces of high
dimension, $\rm{dim}\;{\cal H}>>1.$

If the “conscious-experience vocabulary” $X$ (for observable A) is not so
large, i.e., number of points in set $X<<\rm{dim}\;{\cal H},$ the same
conscious experience $x$ is generated by huge variety of unconscious states.
This degeneration is not good for cognitive behavior - reactions to external
and internal stimuli and communications with other humans, especially for the
latter.

How can the brain, as the self-observable, reduce this mental state
degeneration? The answer is known from quantum theory. Consciousness ${\cal
C}$ has to complete the $A$-observation, $A\in O_{\rm{bs}},$ with the
observation of a compatible observable $B\in O_{\rm{sup}}.$ The latter plays
the role of context for the $A$-observation. The value $A=x$ is contextualized
with the value $B=y.$ In each outcome $(x,y)$ of the joint measurement of
$(A,B),$ the value $x$ represents the basic conscious experience and $y$ its
contextual coloring.

The crucial point is that such contextualization-observable $B$ should not
carry conscious meaning which is directly related to the $A$-meaning,
otherwise the joint observation $(A=x,B=y)$ can essentially modify the meaning
of the outcome $A=x.$ It is also preferable that $B$-observation can be
combined not only with $A$-observation, but with observation of any
$A^{\prime}\in O_{\rm{bas}}.$

We proceed with POV-observables; in particular, to enrich selection of
outcomes’ vocabularies. However, the reader use Hermitian operators and
restrict all vocabularies to subsets of the real line. In the operator-
observables framework, the condition of compatibility is formulated simply as
$[A,B]=0$ for any $A\in O_{\rm{bas}}$ and any $B\in O_{\rm{sup}}.$ In our POV-
observables framework, we proceed with the condition

$[E^{A}_{x},E^{B}_{y}]=0\;\mbox{for
observables}\;A=(E^{A}_{x}),B=(E^{B}_{y}),$ (16)

or symbolically

$[O_{\rm{bas}},O_{\rm{sup}}]=0.$ (17)

In general two observables $A,A^{\prime}\in O_{\rm{bas}}$ do not commute,
i.e., there can exist two projectors such that
$[E^{A}_{x},E^{A^{\prime}}_{x^{\prime}}]\not=0.$ Thus, an arbitrary
$A^{\prime}\in O_{\rm{bas}}$ cannot be used for refinement of $A\in
O_{\rm{bas}}.$

As was mentioned, for the outcome $A=x$ the co-outcome $B=y$ can be considered
as coloring of the experience $A=x.$ Cognitive $O_{\rm{bas}}$-representation
can be compared with black-white pictures of houses in a town, the
$O_{\rm{sup}}$-representation adds colors: the house $A=x$ is “colored” with
the color $B=y.$ (Here we use “coloring” metaphorically.) Conscious
experiences of $(x,y)$ and $(x,\tilde{y}),$ where $x\in X$ and $y,\tilde{y}\in
Y$ (the value-sets for $A$ and $B)$ can differ essentially.

Thus, appeal to supplementary conscious experiences given by the set of
observables $O_{\rm{sup}}$ enriches tremendously the set of basic conscious
experiences. At the level of mental states, it makes correspondence between
states and experiences less degenerate. The latter helps a lot in social
communication between individuals.

In principle, $O_{\rm{bas}}$ also can contain compatible observables, say
$A,A^{\prime}$ such that $[A,A^{\prime}]=0.$ In such a case, the outcomes of
$A^{\prime}$ might be used by ${\cal C}$ for “coloring” of the outcomes of $A$
and vice verse. However, the $O_{\rm{sup}}$-coloring is preferable. The set of
observables $O_{\rm{sup}}$ is specified by ${\cal C}$ at the level of hardware
and soft-ware; ${\cal C}$ need not to check whether an observable from
$O_{\rm{sup}}$ can be used or not for coloring of an arbitrary observable
$A\in O_{\rm{bas}}.$ Another problem with mutual coloring of observables
$A,A^{\prime}$ is that both carry important cognitive meanings and coloring of
$A$ by $A^{\prime}$ (or vice versa) can modify the cognitive meaning of $A,$
since ${\cal C}$ should process simultaneously two basic conscious functions.

And finally, we repeat that generation of outcomes of observables belonging to
$O_{\rm{bas}}$ is slower (in some situations, e.g., for emotional coloring,
essentially slower [31]) than generation of $O_{\rm{sup}}$-observables. So, by
attempting to color $A\in O_{\rm{bas}}$ with $A^{\prime}\in
O_{\rm{bas}},\;{\cal C}$ would consume more time and even time scale
inconsistency can be a problem.

###  8.1 Emotional coloring of perceptions

In the above scheme, we set $O_{\rm{bas}}\equiv O_{\rm{per}}$ and
$O_{\rm{sup}}\equiv O_{\rm{em}}.$

As was stated in section 4.3, the information processing system ${\cal UC}$
contains the following two subsystems. One is involved in processing of
sensations into perceptions – still unconscious processing; denote it as
${\cal UC}_{\rm{per}}.$ Another system ${\cal UC}_{\rm{em}}$ processes
unconscious emotional states. The corresponding state spaces denote by the
symbols ${\cal H}_{\rm{per}}\equiv{\cal H}_{{\cal UC};\rm{per}}$ and ${\cal
H}_{\rm{em}}\equiv{\cal H}_{{\cal UC};\rm{em}},$ respectively.

Generally ${\cal UC}$ is not reduced to ${\cal UC}_{\rm{per}}$ and ${\cal
UC}_{\rm{em}}.$ But, for simplicity, for modeling emotional coloring of
perceptions, we assume that ${\cal UC}$ is compound solely of these
subsystems. This is really oversimplified picture of functioning of
unconsciousness which is not reduced to transformation of sensations into
perceptions and generation of emotional states. But, we proceed in this
framework.

##  9 Tensor product formalism

Now we turn again to the general scheme of modeling of basic and supplementary
conscious experiences. Before we have operated with the corresponding sets of
observables $O_{\rm{bas}}$ and $O_{\rm{sup}},$ now we would like to proceed in
the dual framework by operating with states.

In quantum information theory, contextuality is typically presented within the
tensor product formalism.666 In particular, it is closely coupled to the
experimental tests of contextuality - with the Bell-type inequalities. In
fact, the direct experimental testing of contextuality in the sense of
Definition 1 is impossible. The Bell test is about contextuality of possible
hidden variables modedel beyond quantum observables. In physics, the
contextuality issue is mixed with nonlocality, up to consideration of spooky
action at a distance. Our quantum-like model does not reflect genuine quantum
effects in the brain. Information processing is based on classical
electromagnetic fields (cf. with applications of classical optics in quantum
information theory [30]). Contextuality is a consequence of existence of
incompatible observables, in the spirit of articles [76, 77].

Let, as above, ${\cal H}$ be the state space of ${\cal UC}.$ Suppose that it
is factorized in the tensor product ${\cal H}={\cal H}_{\rm{bas}}\otimes{\cal
H}_{\rm{sup}},$ where ${\cal H}_{\rm{bas}}$ and ${\cal H}_{\rm{sup}}$ are
state spaces coupled to basic and supplementary conscious experiences which
are represented at the conscious level by observables belonging to the sets
$O_{\rm{bas}}$ and $O_{\rm{sup}}.$ These states spaces are generated by two
different unconscious information processing systems, say ${\cal
UC}_{\rm{bas}}$ and ${\cal UC}_{\rm{sup}}.$ These systems are concentrated in
different brain’s areas created at the different stages of the brain’s
evolution. However, concentration is not sharp, processing is distributed and
the processing areas have overlap. The complex Hilbert space ${\cal H}$ is the
state space of the compound system $({\cal UC}_{\rm{bas}},{\cal
UC}_{\rm{sup}}).$ In the present model, it can be identified with
unconsciousness, ${\cal UC}.$ (Generally ${\cal UC}$ has more complex
structure.)

Observables belonging to $O_{\rm{bas}}$ and $O_{\rm{sup}}$ represent
measurements which are performed by ${\cal C}$ on the subsystems ${\cal
UC}_{\rm{bas}}$ and ${\cal UC}_{\rm{sup}},$ respectively. If observables are
mathematically described as Hermitian operators, then each operator $A:{\cal
H}_{\rm{bas}}\to{\cal H}_{\rm{bas}}$ is represented as the operator ${\bf
A}=A\otimes I:{\cal H}\to{\cal H}$ and each operator $B:{\cal
H}_{\rm{sup}}\to{\cal H}_{\rm{sup}}$ is represented as the operator ${\bf
B}=I\otimes B:{\cal H}\to{\cal H}.$ For POV-observables, we use the same
procedure: for $A=(E^{A}_{x})$ and $B=(E^{B}_{y}),$ we set ${\bf
A}=(E^{A}_{x}\otimes I)$ and ${\bf B}=(I\otimes E^{B}_{y}).$ The joint
measurement of two POV-observables is represented by POV $A\otimes
B=(E^{A}_{x}\otimes E^{B}_{y}).$

The state space ${\cal H}={\cal H}_{\rm{bas}}\otimes{\cal H}_{\rm{sup}}$ is
generated by tensor products of the form
$\psi_{\rm{bas}}\otimes\psi_{\rm{sup}}.$ Measurements of observables on such
separable states are reduced to independent measurements on the subsystems
${\cal UC}_{\rm{bas}}$ and ${\cal UC}_{\rm{sup}}$ (for $A,B$ of aforementioned
classes). The real quantum information effects become visible for non-
separable, entangled, states, those states which cannot be represented in the
form of the tensor product.

For example, let both state spaces be two dimensional (qubit spaces), and let
$|j_{a}\rangle$ and $|j_{b}\rangle,j_{a},j_{b}=0,1,$ be orthonormal bases in
${\cal H}_{\rm{bas}}$ and ${\cal H}_{\rm{sup}},$ let observables $A$ and $B$
represented by one dimensional POV corresponding to these bases, i.e.,
$E^{A}_{j_{a}}=|j_{a}\rangle\langle j_{a}|$ and
$E^{B}_{j_{b}}=|j_{b}\rangle\langle j_{b}|,$ where $j_{a},j_{b}=0,1.$ Set
$|j_{a}j_{b}\rangle=|j_{a}\rangle\otimes|j_{b}\rangle;$ this is an orthonormal
basis in the state space ${\cal H}$ of the compound system $S=({\cal
UC}_{\rm{bas}},{\cal UC}_{\rm{sup}}).$ Then the states of $S$ can be expanded
w.r.t. this basis.

$\psi=\sum_{ji}c_{ji}|ji\rangle,\;\sum_{ji}|c_{ji}|^{2}=1.$ (18)

By the Born’s rule $p_{\psi}(A=a_{j},B=b_{i})=|c_{ji}|^{2}$ is the probability
that ${\cal C}$ observes the perception-emotion pair $(a_{j},b_{i}).$

For example, the state

$\psi=(|00\rangle+|11\rangle)/\sqrt{2}$ (19)

is entangled. This state illustrate the correlation meaning of entanglement -
in fact, the maximal entanglement. Suppose that observables $A$ and $B$ yield
the real values $a_{0},a_{1}$ and $b_{0},b_{1},$ respectively. Consider
quantum-like modeling of emotional coloring of perception, i.e., for ${\cal
H}_{\rm{bas}}={\cal H}_{\rm{per}},{\cal H}_{\rm{sup}}={\cal H}_{\rm{em}}.$
Perception $A=a_{j}$ is firmly associated with emotion $B=b_{j},j=0,1.$ Thus,
this state represents the perfect correlations between the $A$-perception and
the $B$-emotion.

For example, consider Russia or France in 19th century, decision maker was an
officer who was participating in some social event and conflicting with
another officer; $A$ is the decision observable, “to challenge $(A=a_{1})$ or
not $(A=a_{0})$ to a duel”; $B$ the emotion observable, “angry $(B=b_{1})$ or
not $(B=b_{0}).$ If an officer is in the (unconscious) mental state (19), then
the emotion “angry” matches perfectly with challenge to a duel.

The general two qubit state given by superposition (18) encodes correlation

$\langle AB\rangle_{\psi}\equiv\langle{\bf A}{\bf
B}\psi,\psi\rangle=\sum_{ji}a_{j}b_{i}|c_{ji}|^{2}=\sum_{ji}a_{j}b_{i}p_{\psi}(A=a_{j},B=b_{i}),$
(20)

w.r.t. state $\psi\in{\cal H}.$ The last sum is the classical probabilistic
expression for correlation. Thus, each concrete pair (perception, emotion) or
(decision, emotion) can be described in the classical probabilistic framework.
In particular, by experimenting with just one pair we would detect quantum-
like effects.

Consider, for the same perception $A\in O_{\rm{per}},$ another emotion
$B^{\prime}\in O_{\rm{em}}$ and correlation $\langle
AB^{\prime}\rangle_{\psi};$ it can be expressed with the coefficients with
respect to the basic
$(|j_{a}j_{b^{\prime}}\rangle=|j_{a}\rangle\otimes|j_{b^{\prime}}\rangle),$

$\langle AB^{\prime}\rangle_{\psi}\equiv\langle{\bf A}{\bf
B}^{\prime}\psi,\psi\rangle=\sum_{ji}a_{j}b_{i}^{\prime}|c_{ji}^{\prime}|^{2}=\sum_{ji}a_{j}b_{i}^{\prime}p_{\psi}(A=a_{j},B^{\prime}=b_{i}^{\prime}).$
(21)

This is also the classical expression for the correlation.

##  10 Bell type inequalities and experimental testing of emotional
contextuality

Now let us consider the correlation-expressions (20), (21) jointly. If
emotional-observables $B=(E^{B}_{y})$ and
$B^{\prime}=(E^{B^{\prime}}_{y^{\prime}}$ are incompatible, i.e., in the
mathematical terms, there exist projectors such that
$[E^{B}_{y},E^{B^{\prime}}_{y^{\prime}}]\not=0,$ then generally it is
impossible to combine these two classical correlations in the single classical
probabilistic model. This is the complex foundational problem which is
formalized with the aid of Bell type inequalities [51, 52, 74]. We are not
able to go deeper into basics of quantum mechanics. We finish with the
following foundational remark. In quantum physics, violation of these
inequalities is coupled to violation of at least one of the followings two
assumptions:

  * •

a) realism, i.e., the possibility to assign the values of observables before
measurement;

  * •

b) locality, the absence of action at a distance.

The combination of a)+b) is known as local realism. In quantum physics, one
does not distinguish the two components of local realism (but, see [76, 77]
for the claim that the key issue is violation of a) and that quantum theory is
local; see also Plotnitsky [78] – [80] for foundational analysis of the
interplay incompatibility-nonlocality). As was argued in section 6, “mental
realism”, i.e., the assumption that say all possible emotions peacefully
coexist in any mental state, is hardly acceptable. Therefore we consider
violation of “mental realism” as the root for violation of the Bell type
inequalities.

In quantum physics, experimental testing of the Bell type inequalities is the
hot topic (see, e.g., [81, 82, 83] for the recent experiments). In psychology
and decision making, they have been tested by a few authors [84] – [88]. This
paper can stimulate such experimenting in consciousness studies with joint
measurements of the pairs $(A,B)=$ (perception, emotion) or (decision making,
emotion).

As in physics and the previous psychological experiments, it is natural to
test the CHSH inequality [89]. To proceed with it, it is not enough to
consider just one perception (decision making) $A$ and two incompatible
emotions $B$ and $B^{\prime}.$ One has to work as well with two incompatible
perceptions $A$ and $A^{\prime}$ and form cyclically their correlations.The
CHSH correlation function is given by the following combination of
correlations:

$C_{\rm{CHSH}}=\langle AB\rangle+\langle AB^{\prime}\rangle+\langle
A^{\prime}B\rangle-\langle A^{\prime}B^{\prime}\rangle$ (22)

and, for dichotomous observables yielding values $\pm 1$ the following
inequality holds:

$|C_{\rm{CHSH}}|\leq 2.$ (23)

Finally, we stress that the psychological studies demonstrated that the humans
behavior differs from the behavior of quantum physical systems w.r.t. so-
called signaling problem [87, 88]. It would be interesting to check whether
experiments with emotions would lead to signaling or not. My conjecture that,
for some mental states, emotional coloring (contextualization) can be
performed without signaling. I stress that the proposed emotion-experiments
differ from the previously performed psychological experiments in which
experimenters operated with pairs composed of two decisions, i.e., one
decision played the role of context for another.

##  11 Concluding remarks

We presented the quantum-like model of emotional coloring of perceptions and
other conscious experiences, including decision making.777We repeat that this
model has no direct coupling to study of genuine quantum physical processes in
the brain. The brain, as the information processor, is decomposed into two
sub-processors, unconsciousness ${\cal UC}$ and consciousness ${\cal C}.$ The
later plays the role of an observer on the former. This is mental realization
of the quantum measurement scheme for self-observatios performed by the brain.
The state space of ${\cal UC}$ is mathematically described as a complex
Hilbert space of very high dimension. In this paper we do not model the
process of ${\cal UC}-{\cal C}$ interaction. Conscious observables are
represented in the operator formalism. Perceptions and emotions are described
by two classes of POV-observables, $O_{\rm{per}}$ and $O_{\rm{em}}.$
Perceptions are compatible with emotions, i.e., they can be jointly observed
by ${\cal C}.$ In the mathematical formalism, compatibility is encoded as
commutativity.

Both perceptions and emotions are treated as conscious experiences. Quantum
measurement formalism matches perfectly to the Higher Order Theory of
Consciousness. Emotions correspond to repeatable contexts and they
contextualize perceptions and other conscious experiences. Emotional coloring
reduces state degeneration for them and makes the information processing less
diffuse. Context-matching is also important for social communication.

One of the main distinguishing features of the quantum measurement theory is
the presence of incompatible, i.e., jointly unobservable entities. In
particular, the presence of incompatible observables makes impossible the use
of the classical probability model (axiomatic of Kolmogorov [90]). The
existence of incompatible perceptions or emotions is evident even from our
personal experience. This motivates the use of the mathematical formalism fo
quantum theory for modeling brain’s self-observations.

To model emotional contextuality, we explore the tensor product formalism by
factorizing the unconscious state space into the tensor product, one of its
components describes the states of emotions. As is well known from quantum
physics, the direct test of contextuality is impossible due to counterfactual
nature of its formulation. The indirect tests of contextuality are based on
the Bell type inequalities. We discuss the possibility of such tests for pairs
(perception, emotion) or (decision making, emotion). As in physics, the
problem of the interpretation of the violation of Bell inequalities is very
complex. Following [76, 77], we couple violation and observables
incompatibility, i.e., the existence of incompatible perceptions and emotions.

We hope that our model can stimulate the applications of the quantum formalism
in conscious-studies, especially in modeling perception-emotion and decision-
emotion correlations.

##  12 Appendix: Commutator representation of CHSH-correlation function

Here we present the purely mathematical result supporting our incompatibility
interpretation of violation of the Bell type inequalities.

Consider Hermitian operators $A,A^{\prime}$ and $B,B^{\prime}.$ In our model,
the first pair belongs to $O_{\rm{per}}$ and the second one to $O_{\rm{em}}.$
The considered perceptions and emotions are assumed to be compatible, i.e.,
$[A,B]=0,...,[A^{\prime},B^{\prime}]=0.$ It is also assumed that the
observables are dichotomous and yield the values $\pm 1.$ In the operator
terms, the latter is expressed as
$A^{2}=(A^{\prime})^{2}=B^{2}=(B^{\prime})^{2}=I,$ where $I$ is the unit
operator.

For quantum observables, the CHSH correlation function $C_{\rm{CHSH}}$ can be
represented as the average of the corresponding Hermitian operators (see [77]
for details):

$C_{\rm{CHSH}}=\langle\Gamma_{\rm{CHSH}}\psi|\psi\rangle,$ (24)

where

$\Gamma_{\rm{CHSH}}=A(B+B^{\prime})+A^{\prime}(B-B^{\prime}).$ (25)

The CHSH inequality has the form:

$|\langle\Gamma_{\rm{CHSH}}\psi|\psi\rangle|\leq 2.$ (26)

By straightforward calculation, one can derive at the Landau identity:

$\Gamma_{\rm{CHSH}}^{2}=4I-[A,A^{\prime}][B,B^{\prime}],$ (27)

where, for two operators $Q_{1},Q_{2},\;[Q_{1},Q_{2}]=Q_{1}Q_{2}-Q_{2}Q_{1}$
is their commutator. Thus, if at least one pair of observables
$(A,A^{\prime})$ or $(B,B^{\prime})$ is compatible, i.e., at least one of
commutators $[A,A^{\prime}],[B,B^{\prime}]$ is equal to zero, then
$\Gamma_{\rm{CHSH}}=2I$ and the CHSH inequality (26) cannot be violated. If
both commutators are nonzero, then, for some state $\psi,$ it can be violated
(see []).

The reader can see that this is purely commutativity-nocommutativity game.

## References

  * [1] J. E. LeDoux and R. Brown, A higher-order theory of emotional consciousness. PNAS, 2017 114 (10) E2016-E2025; https://doi.org/10.1073
  * [2] Khrennikov A. Quantum-like model of unconscious–conscious dynamics. Frontiers in Psychology 6, 2015, 997-1010. https://www.frontiersin.org/article/10.3389/fpsyg.2015.00997
  * [3] Bohr, N. _The Philosophical Writings of Niels Bohr_ ; Ox Bow Press: Woodbridge, UK, 1987.
  * [4] Wigner, E.P., 1961. Remarks on the mind-body question. In: Good, L.J. (Ed.), The Scientist Speculates – An Anthology of Partly Baked Ideas. Heinemann, London. pp. 284-302.
  * [5] Wigner, E.P., 1963. The problem of measurement. Am. J. Phys. 31, 6-15.
  * [6] F. H. Thaheld, Does consciousness really collapse the wave function? A possible objective biophysical resolution of the measurement problem. https://arxiv.org/ftp/quant-ph/papers/0509/0509042.pdf .
  * [7] A.Yu. Khrennikov, Human subconscious as the $p$-adic dynamical system. Journal of Theoretical Biology, 193, 179-196 (1998).
  * [8] D. Dubischar, M. Gundlach, O. Steinkamp, A.Yu. Khrennikov, A $p$-adic model for the process of thinking disturbed by physiological and information noise. Journal of Theoretical Biology, 197, 451-467 (1999).
  * [9] A.Yu. Khrennikov, Information Dynamics in Cognitive, Psychological, Social and Anomalous Phenomena, Springer-Science + Business Media, B.Y., Dordrecht, NL, 2004.
  * [10] Albeverio S, Khrennikov A and Kloeden P E Memory retrieval as a $p$-adic dynamical system BioSystems 49 105–115 (1999).
  * [11] E. A. Liberman, S. V. Minina, N. E. Shklovsky-Kordi, Quantum molecular computer model of the neuron and a pathway to the union of the sciences. Biosystems 22, 1989, 135-154.
  * [12] Penrose, R. (1989). _The Emperor’s new mind_ , Oxford Univ. Press: New-York.
  * [13] Minina S. V., Liberman E. A. Input and output channels of quantum biocomputers. Biofizika. 1990 35(1), 132-136.
  * [14] Igamberdiev, A.U., 1993. Quantum mechanical properties ofbiosystems: a framework for complexity, structural stability andtransformations. BioSystems 31, 65–73.
  * [15] Umezawa, H. (1993). _Advanced field theory: micro, macro and thermal concepts_ , AIP: New York.
  * [16] Hameroff, S. (1994). Quantum coherence in microtubules. A neural basis for emergent consciousness? _J. Cons. Stud._ , 1, 91–118.
  * [17] E.A.Liberman, S.V.Minina, Molecular quantum computer of neuron. Biosystems 35, 1995, 203-207.
  * [18] Vitiello, G. (1995). Dissipation and memory capacity in the quantum brain model, _Int. J. Mod. Phys._ , B9, 973.
  * [19] Vitiello, G. (2001). _My double unveiled: The dissipative quantum model of brain_ , Advances in Consciousness Research, John Benjamins Publishing Company.
  * [20] A. U. Igamberdiev, Quantum computation, non-demolition measurements, and reflective control in living systems. Biosystems 77, 2004, 47-56.
  * [21] Gunji YP, Shinohara S., Haruna T., Basios V. Inverse Bayesian inference as a key of consciousness featuring a macroscopic quantum logical structure. Biosystems. 2017, ;152:44-65.
  * [22] Gunji YP, Sonoda K, Basios V. Quantum cognition based on an ambiguous representation derived from a rough set approximation. Biosystems. 2016 Mar;141:55-66.
  * [23] Khrennikov, A. (2010). _Ubiquitous quantum structure: from psychology to finances_. Berlin-Heidelberg-New York: Springer.
  * [24] Busemeyer, J. & Bruza, P.(2012). Quantum models of cognition and decision. Cambridge: Cambridge Univ. Press.
  * [25] Haven, E. and Khrennikov, A. (2013). _Quantum social science_ ; Cambridge University Press.
  * [26] Asano, M., Khrennikov, A., Ohya, M., Tanaka, Y., Yamato, I. Quantum adaptivity in biology: from genetics to cognition, (Springer, Heidelberg-Berlin-New York, 2015).
  * [27] Haven, E., Khrennikov, A. and Robinson, T. R. (2017). _Quantum Methods in Social Science: A First Course_ ; WSP: Singapore.
  * [28] Bagarello, F. (2019). Quantum concepts in the social, ecological and biological sciences. Cambridge: Cambridge Unive. Press.
  * [29] Khrennikov, A. (2020). Social laser. Jenny Stanford Publ., Singapore.
  * [30] A. Khrennikov, Quantum versus classical entanglement: eliminating the issue of quantum nonlocality. Found Phys 50, 1762–1780 (2020).
  * [31] J. E. LeDoux, Emotional colouration of consciousness: how feelings come about. In: Frontiers of Consciousness. Chichele Lectures. L. Weiskrantz and M. Davies (eds). Oxford Univ. Press, Oxford, 2008.
  * [32] LeDoux, J.E. (1984). Cognition and emotion: processing functions and brain systems. In Gazzaniga, M.S. (ed.) Handbook of Cognitive Neuroscience, pp. 357–368. New York: Plenum.
  * [33] LeDoux, J.E. (1987). Emotion. In Plum, F. (ed.) Handbook of Physiology. 1: The Nervous System. Vol. V, Higher Functions of the Brain, pp. 419–460. Bethesda, MD: American Physiological Society.
  * [34] E. A. Liberman, Analog-digital molecular cell computer. Biosystems 11, 1979, 111-124.
  * [35] M. Conrad, E.A.Liberman, Molecular computing as a link between biological and physical theory. J. Theor. Biology, 98, 1982, 239-252.
  * [36] E. A. Liberman, S. V. Minina, N. E. Shklovski-Kordi, Biological information and laws of nature. Biosystems 46, 1998, 103-106.
  * [37] Liberman, E. A. Cell molecular computer. VII. Cell biophysics and realistic or information physics. Biofizika, 1975, 20(3), 432-436.
  * [38] A. U. Igamberdiev, J. E. Brenner, Mathematics in biological reality: The emergence of natural computation in living systems Biosystems 204, 2021, 104395.
  * [39] Brenner, J. E. Logic in Reality. Dordrecht: Springer, 2008.
  * [40] Brenner, J. E.; A. Igamberdiev, Philosophy in Reality; A New Book of Changes. Berlin-Heidelberg-New York, 2021.
  * [41] Dretske F. (1995) Naturalizing the Mind (MIT Press, Cambridge, MA).
  * [42] Tye M. (2000) Consciousness, Color, and Content (MIT Press, Cambridge, MA).
  * [43] Lamme V. A. F. (2005) Independent neural definitions of visual awareness and attention. Cognitive Penetrability of Perception: Attention, Action, Strategies, and Bottom-Up Constraints, ed Raftopoulos A (Nova Science, New York), pp 171–191
  * [44] Block N. (2011) Perceptual consciousness overflows cognitive access. Trends Cogn. Sci. 15(12):567–575.
  * [45] Block N. (2007) Consciousness, accessibility, and the mesh between psychology and neuroscience. Behav. Brain Sci 30(5-6):481–499, discussion 499–548.
  * [46] Rosenthal DM (2005) Consciousness and Mind (Oxford Univ Press, Oxford).
  * [47] Lau H, Rosenthal D (2011) Empirical support for higher-order theories of conscious awareness. Trends Cogn Sci 15(8):365–373.
  * [48] Carruthers P (2005) Consciousness: Essays from a High-Order Perspective (Oxford Univ Press, Oxford).
  * [49] Von Neumann, J. Mathematical Foundations of Quantum Mechanics; Princeton Univ. Press: Princeton, NJ, USA, 1955.
  * [50] Einstein, A.; Podolsky, B.; Rosen, N. Can quantum-mechanical description of physical reality be considered complete? _Phys. Rev._ 1935, 47, 777–780.
  * [51] Bell, J.S. On the Einstein Podolsky Rosen paradox. Physics 1964, 1, 195–200.
  * [52] Bell, J.S. Speakable and Unspeakable in Quantum Mechanics, 2nd ed.; Cambridge University Press: Cambridge, UK, 2004.
  * [53] Bohr, N. Can quantum-mechanical description of physical reality be considered complete? _Phys. Rev._ 1935, 48, 696–702.
  * [54] James W (1884). What Is an Emotion?. Mind. 9 (34): 188–205.
  * [55] Cannon, W. (1927). The James-Lange Theory of Emotions: A Critical Examination and an Alternative Theory. Amer. J. Psych. 39 (1/4): 106–124.
  * [56] Aziz-Zadeh L, Damasio A (2008). Embodied semantics for actions: findings from functional brain imaging. J. Physiol., Paris. 102 (1–3): 35–9.
  * [57] von Helmholtz, H. (1866). Treatise on Physiological Optics. Transl. by Optical Society of America in English. New York, NY: Optical Society of America.
  * [58] Mesulam M. M. From sensation to cognition. Brain. 1998, 121:1013-52.
  * [59] Ekman P (1992). An argument for basic emotions. Cognition and Emotion. 6 (3): 169–200.
  * [60] James, W. (1890). _The Principles of Psychology_ (New York: Henry Holt and Co.), Reprinted 1983 (Boston: Harvard Univ. Press).
  * [61] Freud, S. (1957). _The Standard Edition of Complete Psychological Works of Sigmund Freud_ , Edited and Translated by J. Strachey, Vols. I-XXIV (The Hogarth Press, London).
  * [62] Jung, C. G. (2001). _On the Nature of the Psyche_ , Routledge Classics.
  * [63] Davies, E. B. Quantum theory of open systems, (Academic Press, London, 1976).
  * [64] Ozawa, M. Quantum perfect correlations. An. Phys. 321, 744-769 (2006).
  * [65] Ozawa, M. Probabilistic interpretation of quantum theory. New Generation Computing 34, 125-152 (2016).
  * [66] Ozawa, M. & Khrennikov, A. (2020). Application of theory of quantum instruments to psychology: Combination of question order effect with response replicability effect. Entropy, 22(1), 37. 1-9436.
  * [67] Atmanspacher, H. Determinism is ontic, determinability is epistemic, in H. Atmanspacher and R. C. Bishop (eds.), _Between Chance and Choice: Interdisciplinary Perspectives on Determinism_ (Imprint Academic, Thorverton UK), 2002, pp. 49–74.
  * [68] Brenner, J. E. Logic in Reality. Dordrecht: Springer, 2008.
  * [69] Brenner, J. E.; A. Igamberdiev, Philosophy in Reality; A New Book of Changes. Berlin-Heidelberg-New York, 2021.
  * [70] Khrennikov, A. (2004). On quantum-like probabilistic structure of mental information, _Open Systems and Information Dynamics_ 11 (3), 267–275.
  * [71] Busemeyer, J. R., Wang, Z. and Townsend, J. T., (2006). Quantum dynamics of human decision making, _J. Math. Psych._ , 50, 220–241.
  * [72] Wang, Z. & Busemeyer, J. R. (2013). A quantum question order model supported by empirical tests of an a priori and precise prediction. Topics in Cognitive Science, 5, 689–710.
  * [73] Khrennikov A. (2016). Quantum Bayesianism as the basis of general theory of decision-making. Philosophical transactions. Series A, Mathematical, physical, and engineering sciences, 374(2068), 20150245.
  * [74] Bell, J.S. On the problem of hidden variables in quantum theory. Rev. Mod. Phys. 1966, 38, 450.
  * [75] Khrennikov, A., Basieva, I., Pothos, E. M. and Yamato, I. (2018). Quantum probability in decision making from quantum information representation of neuronal states, _Scientific Reports_ 8, Article number: 16225.
  * [76] A. Khrennikov, Bohr against Bell: complementarity versus nonlocality. Open Phys. 15, 734–73 (2017).
  * [77] A. Khrennikov, Get rid of nonlocality from quantum physics. Entropy, 21(8), 806 (2019).
  * [78] A. Plotnitsky, “Without in any way disturbing the system”: Illuminating the issue of quantum nonlocality. arXiv:1912.03842 [quant-ph].
  * [79] A. Plotnitsky, Reality, Indeterminacy, Probability, and Information in Quantum Theory. Entropy 2020, 22(7), 747; https://doi.org/10.3390/e22070747
  * [80] A. Plotnitsky, The Unavoidable Interaction Between the Object and the Measuring Instruments: Reality, Probability, and Nonlocality in Quantum Physics. Found Phys (2020). https://doi.org/10.1007/s10701-020-00353-5 .
  * [81] Hensen, B.; Bernien, H.; Dreau, A.E.; Reiserer, A.; Kalb, N.; Blok, M.S.; Ruitenberg, J.; Vermeulen, R.F.; Schouten, R.N.; Abellan, C.; et al. Experimental loophole-free violation of a Belli nequality using entangled electron spins separated by 1.3 km. Nature 2015, 526, 682.
  * [82] Giustina, M.; Versteegh, M.A.; Wengerowsky, S.; H.; steiner, J.; Hochrainer, A.; Phelan, K.; Steinlechner, F.; Kofler, J.; Larsson, J.Å; Abellan, C.; et al. A significant-loophole-free test of Bell’s theo-rem with entangled photons. Phys. Rev. Lett. 2015, 115, 250401.
  * [83] Shalm, L.K.; Meyer-Scott, E.; Christensen, B.G.; Bierhorst, P.; Wayne, M.A.; Stevens, M.J.; Gerrits, T.; Glancy, S.; Hamel, D.R.; Allman, M.S.; et al. A strong loophole-free test of local realism. Phys. Rev. Lett. 2015, 115, 2504.
  * [84] Conte, E., Khrennikov, A., Todarello, O. and Federici, A. (2008). A preliminary experimental verification on the possibility of Bell inequality violation in mental states, _Neuroquantology_ , 6(3), 214–221.
  * [85] Asano, M., Khrennikov, A., Ohya, M., Tanaka, Y. and Yamato, I. (2014). Violation of contextual generalization of the Leggett-Garg inequality for recognition of ambiguous figures, _Physica Scripta_ , T163, 014006.
  * [86] Asano, M., Khrennikov, A., Ohya, M., Tanaka, Y. and Yamato, I. (2015). _Quantum adaptivity in biology: from genetics to cognition_ , Springer: Heidelberg-Berlin-New York.
  * [87] Basieva, I., Cervantes, V.H., Dzhafarov, E.N., Khrennikov, A. (2019). True contextuality beats direct influences in human decision making. J. Exp. Psych.: General 148, 1925-1937.
  * [88] Cervantes, V.H., & Dzhafarov, E.N. (2020). Contextuality analysis of impossible figures. Entropy 22:981; doi.org/10.3390/e22090981.
  * [89] Clauser, J.F.; Horne, M.A.; Shimony, A.; Holt, R.A. Proposed experiment to test local hidden-variable theories. Phys. Rev. Lett. 1969, 23, 880.
  * [90] Kolmolgoroff, A. N. (1933). _Grundbegriffe der Wahrscheinlichkeitsrechnung_ , (Springer-Verlag, Berlin);