# Are Colors Quanta of Light for Human Vision? A Quantum Cognition Study of
Visual Perception

Jonito Aerts Arguëlles
Center Leo Apostel for Interdisciplinary Studies,
Vrije Universiteit Brussel, 1050 Brussels, Belgium
E-Mail<EMAIL_ADDRESS>
###### Abstract

We study the phenomenon of categorical perception within the quantum
measurement process. The mechanism underlying this phenomenon consists in
dilating stimuli being perceived to belong to different categories and
contracting stimuli being perceived to belong to the same category. We show
that, due to the naturally different way in determining the distance between
pure states compared to the distance between density states, the phenomenon of
categorical perception is rooted in the structure of the quantum measurement
process itself. We apply our findings to the situation of visual perception of
colors and argue that it is possible to consider colors as light quanta for
human visual perception in a similar way as photons are light quanta for
physical measurements of light frequencies. In our approach we see perception
as a complex encounter between the existing physical reality, the stimuli, and
the reality expected by the perciever, resulting in the experience of the
percepts. We investigate what that means for the situation of two colors,
which we call Light and Dark, given our findings on categorical perception
within the quantum measurement process.

Keywords:human vision, categorical perception, quantum measurement, Bloch
sphere, quantisation, basic colors, qubits, quantum cognition

##  1 Introduction

We will argue that colors are quanta of light for human vision in a similar
way that photons are quanta of light measured by a physical measuring device.
To gather evidence for this argument, we will draw on various results obtained
over the years within the research field of quantum cognition (Aerts & Aerts,
1995; Aerts & Gabora, 2005b; Busemeyer et al., 2006; Aerts, 2009a, b; Bruza &
Gabora, 2009; Aerts & Sozzo, 2011; Aerts et al., 2012; Busemeyer & Bruza,
2012; Haven & Khrennikov, 2013; Khrennikov, 2014; Dalla Chiara et al., 2015;
Pothos et al., 2015; Blutner & beim Graben, 2016; Moreira & Wichert, 2016;
Yearsley, 2017; Aerts Arguëlles, 2018; Busemeyer et al., 2019; Surov et al.,
2019; Aerts Arguëlles & Sozzo, 2020; Aerts & Beltran, 2022a), some of which
also involve our own work (Aerts Arguëlles, 2018; Aerts Arguëlles & Sozzo,
2020). Nevertheless, we very deliberately want to make this article
understandable to those who have not studied the investigations on which it
relies, so that in addition to references to this research, and more technical
portions of the article, we will also represent in an intuitive clear way the
core aspects in which we frame our claim.

Our argument rests on a fundamental property of human perception, which is not
limited specifically to human vision, but is present in all forms of human
perception, and which has its origin in a phenomenon called ‘categorical
perception’ (Harnad, 1987; Goldstone & Hendrickson, 2010; Aerts & Aerts
Arguëlles, 2022). Categorical perception ‘introduces a warping’ of the
‘stimulus’ that generated the perception in a very specific way. Namely,
stimuli belonging to the same category are perceived in a way that they are
more similar for the one perceiving than an independent of human bias
measurement of the stimuli reveals. Additionally, for stimuli belonging to a
different category, the reverse warping takes place, they are perceived in a
way that they are more different for the one perceiving than an independent of
human bias measurement of the stimuli reveals. There is a long history (Bruner
& Postman, 1949; Liberman et al., 1957, 1967; Lane, 1965; Eimas et al., 1971;
Lawrence, 1949; Berlin & Kay, 1969; Davies et al., 1998; Davidoff, 2001;
Regier & Kay, 2009; Havy & Waxman, 2016; Hess et al., 2009; Disa et al., 2011;
Sidman, 1994; Schusterman et al., 2000; Rosch, 1973; Collier at al., 1973) in
which the phenomenon of categorical perception was identified incrementally,
and in Section 2 we give a brief description of this genesis.

In the reasoning we develop we also rely on a detailed study of the quantum
measurement process, and more specifically on the way this was analyzed, using
a specific measurement model, by our research group in the foundations of
quantum mechanics at the Free University of Brussels (Aerts, 1986; Aerts &
Sassoli de Bianchi, 2015, 2016, 2017). The mathematical model we developed is
an extension of the Bloch model, where not only the pure states, the collapse
probabilities and the decoherenced density states are represented, but also
the measurements and the changes caused by them. We will see in the course of
our analysis how this extended Bloch model is a fertile ground for identifying
the phenomenon of categorical perception, and we will show that the quantum
measurement model already intrinsically contains the structure that gives rise
to the presence of the measurement bias of categorical perception. We will
then gather the arguments that allow us to consider within the analysis we
propose the basic colors as quanta of light for human visual perception in a
similar way that photons are considered quanta of light for physical
measurement apparatuses. For reasons of simplicity we introduce the
mathematical elements for our analysis for the case of two colors that we call
Light and Dark, mentioning that fundamentally the same analysis can be made
for multiple colors with mathematical elements that then belong to a more than
two dimensional complex Hilbert space. No other specific difficulties appear
in this more dimensional situation that are not encountered in the two
dimensional case, namely that of two colors and a two dimensional Hilbert
space, which we consider in detail in the present article.

In Section 2 we give a brief overview of the phenomenon of categorical
perception with specific attention to the case of colors. In Section 3 we
introduce in a detailed and self-prescriptive way, so that prior knowledge is
not necessary, the quantum measurement model, paying attention to the
extension of the traditional Bloch model that also includes a representation
of the measurement and the changes caused by it. In Section 4, we analyze the
way the quantum measurement changes distances between states and show that
this way contains exactly the warping associated with categorical perception.
In other words, the bias caused by categorical perception is intrinsically
contained in the structure of the quantum measurement process. In Section 5 we
gather the elements of the previous sections to give expression to our claim
that we can consider colors as quanta of light for human visual perception.

##  2 Categorical Perception, Quantization and Colors

In this section we wish to bring out the phenomenon of categorical perception,
because the nature of this phenomenon, along with the details of the
measurement model of quantum mechanics, which our next Section 3 deals with,
plays a fundamental role in our claim that colors are quanta of light for
human vision. It is interesting to reflect for a moment on the emergence of
what was later commonly called categorical perception because it illustrates
well the role that certain goals and hypotheses can play.

Around 1950, research on speech perception came into focus as work was being
done on a speech device, the ‘pattern playback machine’, which was intended to
make it possible to automatically convert texts into spoken form so that, for
example, blind people could read with it. This required analyzing speech very
thoroughly, and it is within this setting that Alvin Liberman identified the
phenomenon that would become known as categorical perception. More
specifically, he noticed that when generating a continuum of evenly
distributed consonant-vowel syllables with endpoints reliably identified as
‘b’, ‘d’ and ‘g’, there is a point of rapid decrease in the probability of
hearing the sound as a ‘b’ to hearing it as ‘d’. At a later point, there is a
rapid switch from ‘d’ to ‘g’ (Liberman et al., 1957). Liberman formulated an
original hypothesis by which he wished to explain why people perceive an
abrupt change between ‘b’ and ‘p’ in the way speech sounds are heard in
contrast to what happens with a synthetic morphing device that produces the
sounds with a continuous transition. His hypothesis was that this phenomenon
is due to a limitation of the human speech apparatus which, due to the
muscular nature of its construction, would be unable to produce continuous
transitions. Because of the way people produce these sounds as they speak,
people’s natural vocal apparatus would be unable to pronounce anything between
‘b’ and ‘p’. So when someone hears a sound from the synthetic morphing device,
that person tries to compare that sound with what he or she would have to do
with his or her voice device to produce this sound. Since a human tuning
device can only produce ‘b’ or ‘p’, all continuous synthetic stimuli will be
perceived as ‘b’ or ‘p’, whichever is closest.

The hypothesis was also the basis of what is now called the ‘motor theory of
speech perception’, which assumes that people perceive spoken words by
recognizing the gestures in the speech channel used to pronounce them, rather
than by identifying the sound patterns that produce the speech (Liberman et
al., 1967). The theory came under fire when it was found that ‘identification’
and ‘discrimination’ of stimuli not at all associated with speech behave in a
similar way to stimuli associated with speech when measured in a similar
manner (Lane, 1965). It was also found that children, even before they could
speak, exhibited the specific categorical perception effect associated with
speech perception that had been identified in adults (Eimas et al., 1971).

So, step by step it became clear that categorical perception was a much more
general phenomenon than just associated with speech, and the connection was
made with earlier findings having to do with the way stimuli are organized. In
particular, Lawrence’s experiments and his hypothesis of ‘acquired
distinctiveness’ revealed a phenomenon that turned out to be a very basic
effect of perception. The hypothesis of acquired distinctiveness states that
stimuli for which one is taught to give a different response to them become
more distinctive, while stimuli for which one is taught to give the same
response to them become more similar (Lawrence, 1949). Both effects are at
work in humans in a multitude of perceptions, stimuli that fall within the
same category are perceived as more similar, while stimuli that fall into
different categories are perceived as more different. What happens with
‘colors’ is a good example of the phenomenon of categorical perception. We see
a discrete set of colors while from physical reality there is a continuum of
different frequencies presented to us as stimuli. The warping effect of the
categorical perception of colors consists in two stimuli that both fall within
the category of, for example, green to be perceived more equally than two
stimuli one of which falls within the category of green and the second within
the category of blue, even if from a physical perspective both pairs of
stimuli have the same difference in frequency. The cooperation of these two
effects, a contraction within an existing category and a dilation between
different categories, causes a clumping of colors, ultimately leading to the
colors that we distinguish.

Given the research on colors that we present in this article, it is
appropriate to mention Eleanor Rosch’s work on colors. Indeed, it was that
work that inspired her to propose the prototype theory of concepts, which
remains one of the most important theories of concepts today (Rosch, 1973).
The basic idea of prototype theory is that there exists a central element for
a concept, which we call the prototype, relative to which the exemplars of the
concept can be placed within a graded structure. Rosch suggested the idea that
would develop into the primary model for concepts by studying the categorical
structure among the Dani for colors and basic shapes. The Dani are a people
living in Papua New Guinea, with the peculiarity that they have only two words
to denote colors, one meaning Bright and the other Dark 111As in other
articles of our Brussels research group, we denote concepts, when they appear
as subjects in the text, by writing them beginning with a capital letter and
in italic.. The Dani also have no words in their language for basic shapes
such as Circle, Square and Triangle. Rosch examined whether there was a
difference in learning between two groups of Dani volunteers, with one group
learning colors and basic shapes, starting with stimuli that are prototype
colors and prototype basic shapes, while the other group learned to start with
stimuli that are different distortions of these prototypes. In a significant
way, it was found that for both colors and basic shapes, learning was more
qualitative for the group taught starting from the prototype stimuli. This
evaluation of ‘more qualitative learning’ took into account the three features
of how this can be measured, namely by the ease of learning sets of shape
categories when a particular type was the prototype, by the ease of learning
individual types within sets, and by rank order of rating types as the best
example of categories, when the prototypes of both colors and shapes were the
stimuli in the learning process.

Mathematical prototype models based on fuzzy sets were developed for concepts
and experimentally tested, and it seemed that the way in which by warping,
i.e., contraction when stimuli fall into the same category and dilation when
they fall into different categories, concepts emerge and grow from stimuli,
had finally found a form to understand what is taking place at a fundamental
theoretical level (Collier at al., 1973; Rosch, 1975; Rosch et al., 1976;
Smith & Medin, 1981; Medin at al., 1984; Geeraerts et al., 2001; Johanden &
Kruschke, 2005).

A fundamentally not understood problem however, even when good mathematical
models existed by which a concept and its set of exemplars and features could
be modeled according to the approach of prototype theory, was the description
of the combination of two concepts. This problem was noted in a first
publication in which the combination of the concept Pet with the concept Fish
served as an example, and therefore the problem of combining two concepts is
often called the ‘pet-fish problem’, or also the ‘guppy effect’, because Guppy
was the exemplar used to illustrate what goes wrong with prototype theory when
concepts are combined (Osherson & Smith, 1981). The question posed was ‘how is
it that Guppy is not a typical example of a Pet, nor a typical example of a
Fish, but a very typical example of a Pet-Fish’. It is starting from this
‘guppy effect’ that in our Brussels research group it was tried successfully
whether a quantum formalism could be used to model the guppy effect
mathematically, where then the sudden relevance of Guppy as a typical exemplar
would be explained as an ‘interference effect’ between the concepts Pet and
Fish when combined (Aerts & Gabora, 2005a, b). These first models using the
quantum formalism were further developed and refined in the following years
(Aerts, 2009a, b) and their relevance to quantum information science was
demonstrated by also modeling data obtained from the World-Wide Web in a
similar way with a quantum model (Aerts et al., 2012).

Meanwhile, it became clear that there were many more similarities between the
structure and dynamics to which concepts of human language are subject and the
structure and dynamics of quantum entities as described by quantum mechanics.
The well-studied phenomenon of quantum entanglement, for example, also occurs
with concepts, it is possible to combine concepts in a very similar way as is
the case with quantum entities, such that experiments on these combinations
violate Bell’s inequalities, which is the experimental test for the presence
of entanglement (Aerts & Sozzo, 2011, 2014). Another more recent finding
consists in identifying the statistical properties of texts of stories, short
stories and stories of the length of novels. It could be shown in a very
convincing way that the statistics inherent in such texts is of the Bose-
Einstein type, hence the same as the statistics of a class of quantum entities
called bosons, such as photons. It could also be shown that it cannot be of
the Maxwell-Boltzmann type – as one would expect it to be, since Maxwell-
Boltzmann is the common statistics for a collection of classical entities
(Aerts & Beltran, 2020). It was also investigated and shown that regarding the
thermodynamic properties of texts representing stories it is the quantum
mechanical von Neumann entropy that is relevant and not the classical Shannon
entropy (Aerts & Beltran, 2022a, b). But the most important finding of a
recent nature as far as the present article is concerned, is the one related
to the phenomenon of quantization. It became clear that the phenomenon of
categorical perception triggers a dynamics that leads to the emergence of
quanta (Aerts & Aerts Arguëlles, 2022). In the present article, we will again
focus on categorical perception and how it gives rise to the formation of
quanta, and more specifically, we will show that the mechanism of categorical
perception is intrinsically present in the structure of the quantum
measurement itself. We will then investigate in what way colors can be
considered to be quanta of light for human vision.

Rosch, during her work on colors, assumed that the way humans see colors is
determined, probably genetically, at birth, also to regard the color
frequencies of the basic colors. More so, that the situation with colors is
thus, was actually her inspiration for proposing the prototype theory for
concepts. This also means that the so-called Sapir-Whorf hypothesis, namely
that there is an influence of ‘how we name categories within a language’ and
‘the perception of stimuli belonging to these categories’, was believed to be
not applicable to colors. And this was generally accepted, colors were thought
to be not subject to how they were named in different languages. Not only do
most cultures divide the groups of colors similarly and give them separate
names, but for the few cultures where this is not the case, the areas of
compression and dilation were assumed to be the same. It was believed that we
all see blue and green in the same way, with a blurred area in between, even
if the naming is not the same (Berlin & Kay, 1969). However, this view was
challenged by studies that nevertheless identified effects of the words
designating colors. Comparative research on colors between speakers of
Setswana, a Bantu language spoken by about 8.2 million people in South Africa,
and speakers of English found many similarities, but also identified
differences relevant in terms of the Sapir-Whorf hypothesis (Davies et al.,
1998). Speakers of Berinmo, an indigenous language in Papua New Guinea, have
only one word, ‘nol’, for what English speakers call green and blue. The
difference they made compared to speakers of English in color discrimination
tasks regarding shades between green and blue was investigated and determined
(Davidoff, 2001). Later evidence was also found that linguistic categories
influence categorical perception primarily in the right visual field. Since
the right visual field is controlled by the left hemisphere, this finding was
explained by the fact that language skills are also located in the left
hemisphere (Regier & Kay, 2009).

More recent experiments, meanwhile, have demonstrated very thoroughly that
language and the names given have an influence on the categorization that
takes place over very primitive visual perceptions. Nine-month-old infants
were shown a continua of new creature-like objects. There was a learning phase
in which the infants were shown that objects from one end of the perceptual
continuum moved to the left and objects from the other end moved to the right.
For one group of infants, the objects were always called by the same name,
while for the other group of infants, two different names were used to call
the objects depending on whether they belonged to one end or the other of the
continuum. The test involved showing new objects from the same continuum to
all infants and then seeing if there was a difference between the two groups.
What was found is that infants in the one-name condition formed one
overarching category and looked at new test objects in either place. Infants
in the two-name condition distinguished two categories and correctly
anticipated the likely location of test objects even when they were near the
poles or near the middle of the continuum (Havy & Waxman, 2016).

However, the experiment with the nine-month-old children we describe above,
that demonstrates the effective existence of a Sapir-Whorf effect with colors,
is not in contradiction with the findings that inspired Rosch. The colors we
customarily distinguish do exist at birth, but the way these colors will play
a role dynamically in an individual person’s life, for this more advanced
function of what colors are, that is where a Sapir-Whorf effect does play a
specific role. As we shall see further in the course of this article, this is
the situation that is captured if colors are considered to be quanta of light
for human visual perception, indeed, it is then the quantum mechanical
structures of superposition and entanglement that can account for these
effects.

In her experiments with the Dani, Rosch used the basic colors as defined by
Brent Berlin and Paul Kay in their authoritative work ‘Basic Color Terms:
Their Universality and Evolution’ (Berlin & Kay, 1969), and we will use these
colors as basic colors in this article as well. By the way, let us note that
these are not the colors of the rainbow, Berlin & Kay (1969) suggested the
following eleven colors as basic colors within the culture where English is
used as a native language, White, Black, Red, Yellow, Green, Blue, Brown,
Purple, Pink, Orange and Gray. Berlin and Kay’s work studies basic color
naming in different language regions and concludes that seven stages in naming
these basic colors can be distinguished from an evolutionary perspective. In
the first stage, only two colors are named, White (then also meaning Light)
and Black (then also meaning Dark). In a second stage, where three names for
basic colors exist, Red is systematically added. To the next two stages,
Yellow and Green are introduced, sometimes Yellow first and sometimes Green
first. At the next stage, Blue is added, and Brown joins the names of basic
colors at the stage after that. We have then reached the seventh stage where
eight to eleven names exist for basic colors, the names still added being
Purple, Pink, Orange and Gray.

The color system used both by Berlin and Kay and by Rosch, and in which they
thus define the eleven prototype colors belonging to the English language, is
the Munsell color system, a three-dimensional system where the sizes of the
three parameters, hue, chroma and value define each color. Berlin and Kay use
their original nomenclature of ‘focus colors’ for the prototype colors, and
Rosch also uses this nomenclature in her earliest articles on colors (Rosch
Heider, 1971, 1972; Mervis et al., 1975). Since it was mainly her research on
colors that made Rosch develop the prototype theory for concepts, she calls
these focus colors in her later work the prototype colors and we will use this
terminology in our writing on colors.

In the next section we examine the connection between categorical perception
and quantum measurement, more specifically, we look in detail at the situation
of a two-dimensional quantum entity and show how its measurement dynamics is
the underlying ground for the warping mechanism at work in the phenomenon of
categorical perception.

##  3 Quantum Measurement and the Bloch Model

We will work with the Bloch representation of a two-dimensional quantum entity
that we will analyze with the intention of identifying the mechanism of
categorical perception at work. Traditionally, the Bloch model is only a
representation of the set of quantum states, pure states and density states,
also called mixed states, of a two-dimensional quantum entity, however, in the
1980s an extension of the Bloch model was worked out in which also the
measurements could explicitly be represented (Aerts, 1986). In later years the
original extension was elaborated and perfected (Aerts & Sassoli de Bianchi,
2014, 2016), and since this extension contains also a specification of the
change provoked by the measuring apparatus on the state of the measured upon
entity, we will use it for our analysis, because it will more easily allow us
to identify the mechanism of categorical perception. We introduce the extended
Bloch model for a two-dimensional quantum entity step by step, explaining
details as they arise, the intention being to make the content of our article
accessible to readers who are not experts in quantum physics. In parallel, we
also introduce the elements of the traditional complex Hilbert space formalism
of a two-dimensional quantum entity. That we speak of a ‘two-dimensional’
quantum entity, by the way, refers to that its Hilbert space has dimension 2.
Within the discipline of quantum computation a two-dimensional quantum entity
is called a ‘qubit’.

A two-dimensional complex Hilbert space $\mathcal{H}$ is a vector space of
dimension 2 over the complex numbers with an inner product, which is a map,
conjugate linear in its first variable and linear in its second, to the set
${\mathbb{C}}$ of complex numbers

$\displaystyle\langle\ |\
\rangle:\mathcal{H}\times\mathcal{H}\rightarrow{\mathbb{C}}$ (1)
$\displaystyle\langle
a\psi_{1}+b\psi_{2}|\psi\rangle=a^{*}\langle\psi_{1}|\psi\rangle+b^{*}\langle\psi_{2}|\psi\rangle$
(2)
$\displaystyle\langle\psi|a\psi_{1}+b\psi_{2}\rangle=a\langle\psi|\psi_{1}\rangle+b\langle\psi|\psi_{2}\rangle$
(3)
$\displaystyle\langle\psi_{1}|\psi_{2}\rangle=\langle\psi_{2}|\psi_{1}\rangle^{*}$
(4)

The inner product allows to express when two states are orthogonal to each
other, namely, if their inner product is equal to zero. It also introduces a
norm on the Hilbert space, the inner product of a vector with itself is the
square of the length of this vector.

In Figures 1 and 2 we consider the basis structure of the Bloch model for a
two-dimensional quantum entity, namely the Bloch sphere, which is a sphere in
three-dimensional Euclidean space with radius equal to 1. The points of the
surface of this sphere represent the pure states of the considered quantum
entity. These are the states that the entity can occupy independently of being
measured upon or not, thus they describe the reality of the considered entity.
In Hilbert space, these pure states are represented by unit vectors of the
Hilbert space. We promised that we wanted to keep the content of this article
understandable for those who do not know the mathematical technicalities of
the quantum formalism, and we will do our best to keep this promise.
Nevertheless, we are obliged to introduce some mathematical elements that are
typical of the quantum formalism. The pure states of two dimensional quantum
entity, hence a qubit, or the spin of a quantum particle with half integer
spin, are represented in the extended Bloch model by the points of the surface
of the Bloch sphere.

Figure 1: A representation of our extended Bloch model for a two-dimensional
quantum entity. A little ball is placed in a point $A$ on the surface of a
sphere and this represents the state of the entity in $A$ before the
measurement. On a central axis of the sphere between the points $A_{down}$ and
$A_{up}$ situated on the surface of the sphere is placed an elastic and a
first phase of the measurement consists of the ball falling through the
interior of the sphere orthogonally to the elastic and sticking to it in a
point $A^{\prime}$ on the centerline between $A_{down}$ and $A_{up}$. The
second phase of the measurement consists of the elastic breaking in one of its
points randomly and uniformly, hence such that the probability of breaking in
a piece of the elastic is proportional to the length of this piece. A a
consequence the ball is moved upwards ending up in the point $A_{up}$ or
downwards ending up in the point $A_{down}$ depending on whether the elastic
breaks in the part down to $A^{\prime}$ or the part up to $A^{\prime}$.

We will denote these points by vectors in the three-dimensional Euclidean
space, indeed, the Bloch sphere is contained in this three-dimensional
Euclidean space, by the so-called spherical coordinates $r$, $\theta$, and
$\phi$. The distance from the center of the Bloch sphere to a point on the
surface of the Bloch sphere is traditionally denoted in spherical coordinates
by $r$, and called the ‘radial distance’. Since our Bloch sphere has radius
equal to 1, for any point on the surface we have $r=1$. The angle that the
line connecting the center of the Bloch sphere with the considered point of
the surface of the Bloch sphere with the axis of the Bloch sphere is denoted
by $\theta$, and it thus varies between values $0$ and $\pi$, it is equal to
$0$ if the considered point coincides with the North Pole of the Bloch sphere
and is called the polar angle (see Figure 2). Hence, it is equal to $\pi$ if
it coincides with the South Pole 222We denote the magnitude of angles by the
unit of a radian, which makes 180 degrees equal to $\pi$ radians. A second
angle is still needed so that the values of $r$, and $\theta$, and the value
of that second angle, traditionally denoted $\phi$ and called the azimuthal
angle, would uniquely determine each point of the three dimensional Euclidean
space. This angle $\phi$ is chosen from a vertical plane through the axis of
the Bloch sphere rotating around that axis reaching the other points not in
the vertical plane, for the points in the plane $\phi$ is equal to zero or
$2\pi$. This means that for values of $\phi$ varying between $0$ and $2\pi$,
all points of three-dimensional Euclidean space can be reached and hence the
spherical coordinates ($r$, $\theta$, $\phi$) form complete coordination of
the three dimensional Euclidean space. For $r=1$, by varying the values of
$\theta$ and $\phi$, all points of the surface of the Bloch sphere are reached
(see Figures 1 and 2). So the point $A$ has spherical coordinates
$(1,\theta,\phi)$, and cartesian coordinates $(x,y,z)$, where

$\displaystyle x$ $\displaystyle=$ $\displaystyle\sin\theta\sin\phi$ (5)
$\displaystyle y$ $\displaystyle=$ $\displaystyle\sin\theta\cos\phi$ (6)
$\displaystyle z$ $\displaystyle=$ $\displaystyle\cos\theta$ (7)

for a cartesian coordinate system with its origin in the center of the sphere.

The coordinates, be they spherical $(r,\theta,\phi)$ or cartesian $(x,y,z)$,
that we have considered so far, describe a three-dimensional Euclidean space,
of which the Bloch sphere is a part.

Figure 2: A three dimensional representation of the Extended Bloch Model. The
little ball is in point $A$ with spherical coordinates $(\rho,\theta,\phi)$
and Hilbert space density operator coordinates $D_{(\rho,\theta,\phi)}$. The
process of measurement proceeds like described in detail in Figure 1 by the
ball falling orthogonally to the elastic and sticking to it in point
$A^{\prime}$. Then the elastic breaks uniformly in one of its points an pulls
the ball upwards to end in $A_{up}$ or downwards to end in $A_{down}$.

However, we have explicitly discussed a two-dimensional quantum entity that we
want to describe using its Bloch representation, while the Cartesian and
spherical coordinates describe a three-dimensional space. If one remembers the
representation of the complex numbers in the Euclidean plane, a mathematics
subject of high school education, then it comes as no surprise that a
representation of a two-dimensional complex vector space needs three
dimensions to be represented in a Euclidean real space. Indeed, the set of
complex numbers, if one wishes to think of it that way, is a one-dimensional
complex vector space and it needs a plane, the so called complex plane, hence
a two-dimensional space, if represented by points with coordinates that are
real numbers, such as is the case by polar coordinates in the complex plane.
Let us introduce a general vector from the two-dimensional vector space, and
we immediately refer to the spherical coordinates to indicate to which point
of the Bloch sphere that vector belongs.

$\displaystyle|\theta,\phi\rangle=(\cos{\theta\over 2}e^{-i{\phi\over
2}},\sin{\theta\over 2}e^{i{\phi\over 2}})$ (8)

We do not yet directly explain here why precisely this vector
$|\theta,\phi\rangle$ of the two-dimensional complex Hilbert space is the one
we let correspond to that point $(1,\theta,\phi)$ of the three-dimensional
Euclidean space, the analysis we make in the course of this article will
clarify this.

The notation $|\theta,\phi\rangle$ is the one introduced by Paul Dirac, one of
the founding fathers of quantum mechanics, along with Werner Heisenberg and
Erwin Schrödinger (Dirac, 1939). Dirac considered the inner product as a
fundamental operational element, the bracket,
$\langle\theta,\phi|\theta,\phi\rangle$, and called the right side of this
bracket, the ket, $|\theta,\phi\rangle$ and the left side, the bra
$\langle\theta,\phi|$. Hence the notation $|\theta,\phi\rangle$ as the right
side, thus as ket, was chosen as a representation of the state of the
considered entity by Dirac because the inner product is taken to be ‘linear’
in the second variable while ‘conjugate linear’ in the first variable 333This
choice, by the way, is sometimes not followed by mathematicians who define the
inner product in a complex Hilbert space as linear in the first variable and
conjugate linear in the second.. In matrix notation, in the case of a two
dimensional quantum entity, the ket is written as a ‘one column’ to ‘two row’
matrix, and the bra as a ‘one row’ to ‘two column’ matrix, plus a complex
conjugation of the terms of the matrix for the bra.

$\displaystyle|\theta,\phi\rangle$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\cos{\theta\over 2}e^{-i{\phi\over 2}}\\\
\sin{\theta\over 2}e^{i{\phi\over 2}}\end{pmatrix}$ (9)
$\displaystyle\langle\theta,\phi|$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\cos{\theta\over 2}e^{i{\phi\over
2}}&\sin{\theta\over 2}e^{-i{\phi\over 2}}\end{pmatrix}$ (10)

We thus find by a direct matrix calculation that the ket is normalized as a
state, indeed we have

$\displaystyle\langle\theta,\phi|\theta,\phi\rangle$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\cos{\theta\over 2}e^{i{\phi\over
2}}&\sin{\theta\over 2}e^{-i{\phi\over
2}}\end{pmatrix}\begin{pmatrix}\cos{\theta\over 2}e^{-i{\phi\over 2}}\\\
\sin{\theta\over 2}e^{i{\phi\over 2}}\end{pmatrix}$ (11) $\displaystyle=$
$\displaystyle\cos^{2}{\theta\over 2}+\sin^{2}{\theta\over 2}=1$

Dirac also introduced in his bra-ket calculus the possibility of multiplying
the ket by the bra in the reverse order of the bra-ket, that is, as ket-bra.
Whereas the bra-ket multiplication always results in a complex number, more
specifically the number 1 in the example we considered here, the reverse
multiplication $|\theta,\phi\rangle\langle\theta,\phi|$ leads to a more
complex quantity. We can calculate this more complex quantity directly in the
case of the two-dimensional quantum entity we are considering, and by the
representations of ket and bra by the two-by-one matrices, as we just
explained. We then get

$\displaystyle|\theta,\phi\rangle\langle\theta,\phi|$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\cos{\theta\over 2}e^{-i{\phi\over 2}}\\\
\sin{\theta\over 2}e^{i{\phi\over
2}}\end{pmatrix}\begin{pmatrix}\cos{\theta\over 2}e^{i{\phi\over
2}}&\sin{\theta\over 2}e^{-i{\phi\over 2}}\end{pmatrix}$ (12) $\displaystyle=$
$\displaystyle\begin{pmatrix}\cos^{2}{\theta\over 2}&\cos{\theta\over
2}\sin{\theta\over 2}e^{-i\phi}\\\ \cos{\theta\over 2}\sin{\theta\over
2}e^{i\phi}&\sin^{2}{\theta\over 2}\end{pmatrix}$

a ‘two-by-two’ matrix. This two-by-two matrix represents an operator within
the quantum Hilbert space formalism called an orthogonal projection, but
before we continue on that, we wish to specify using the extended Bloch model
what happens during a quantum measurement.

Within the extended Bloch model, the measurement is represented by a piece of
elastic stretched on the centerline between two diametrically opposed points
of the sphere, $A_{down}$ and $A_{up}$ (see Figures 1 and 2). On the surface
of the sphere is a small ball that sticks to the sphere at a point $A$. This
ball in a point represents the state in which the quantum entity is. The
measurement then occurs as follows. First, the ball falls orthogonally on the
elastic and sticks to it at the point $A^{\prime}$ where this orthogonal fall
brings it. Then the elastic breaks randomly in one of its points. If the
breaking point is below the point $A^{\prime}$, then the unbroken part of the
elastic pulls the ball up so that it ends up in the point $A_{up}$. However,
if the breaking point is above the point $A^{\prime}$, then the ball is pulled
down by the unbroken part of the elastic, and ends up in the point $A_{down}$.
Thus, the whole of the measurement results in the ball from the point $A$
ending up in one of two points $A_{up}$ or $A_{down}$.

Now suppose that the elastic possesses a uniform breaking pattern, which means
that the point at which the elastic breaks randomly will lie in a certain
interval of the elastic with a probability proportional to the length of that
interval. We can then calculate the probabilities $P(A\mapsto A_{up})$ and
$P(A\mapsto A_{down})$ with which the measurement will carry the ball from $A$
to $A_{up}$ or to $A_{down}$, and in this way will change the state of the
quantum entity, from the simple geometry of the configuration. Note indeed
that we have that the length $L_{2}$ of the elastic under the point
$A^{\prime}$ equals $1+\cos\theta$ and the length $L_{1}$ of the elastic above
the point $A^{\prime}$ equals $1-\cos\theta$, which hence, taking into account
the uniform nature of the breaking pattern of the elastic, gives us

$\displaystyle P(A\mapsto A_{up})$ $\displaystyle=$
$\displaystyle{1+\cos\theta\over 2}=\cos^{2}{\theta\over 2}$ (13)
$\displaystyle P(A\mapsto A_{down})$ $\displaystyle=$
$\displaystyle{1-\cos\theta\over 2}=\sin^{2}{\theta\over 2}$ (14)

The probabilities (13) and (14) match the quantum probabilities of the spin of
a spin 1/2 quantum particle or a qubit. The ball in point $A$ is then in a
state for the spin of an angle $\theta$ with the $z$-axis, and the elastic
lies on this $z$-axis.

The part of this quantum measurement that takes place at the so-called pure
states is described in a complete way in the quantum Hilbert space formalism.
The complex Hilbert space in this case is two-dimensional, with basis vectors
$|0,0\rangle$ and $|\pi,0\rangle$, which in the Bloch representation
correspond to points on the Bloch sphere located at the North and South Poles,
respectively. In Hilbert space these basis vectors, and we use now their
notation as one column and two row matrices, are

$\displaystyle|0,0\rangle=\begin{pmatrix}\cos{\theta\over 2}e^{-i{\phi\over
2}}\\\ \sin{\theta\over 2}e^{i{\phi\over
2}}\end{pmatrix}_{(\theta=0,\phi=0)}=\begin{pmatrix}1\\\ 0\end{pmatrix}$ (15)
$\displaystyle|\pi,0\rangle=\begin{pmatrix}\cos{\theta\over 2}e^{-i{\phi\over
2}}\\\ \sin{\theta\over 2}e^{i{\phi\over
2}}\end{pmatrix}_{(\theta=\pi,\phi=0)}=\begin{pmatrix}0\\\ 1\end{pmatrix}$
(16)

A general pure state $|\theta,\phi\rangle$, in this Hilbert space formalism,
is a normalized linear combination, called superposition in the quantum
jargon, of these basis vectors

$\displaystyle|\theta,\phi\rangle=\begin{pmatrix}\cos{\theta\over
2}e^{-i{\phi\over 2}}\\\ \sin{\theta\over 2}e^{i{\phi\over
2}}\end{pmatrix}=a\begin{pmatrix}1\\\ 0\end{pmatrix}+b\begin{pmatrix}0\\\
1\end{pmatrix}$ (17) $\displaystyle{\rm with}\ a=\cos{\theta\over
2}e^{-i{\phi\over 2}}{\rm and}\ b=\sin{\theta\over 2}e^{i{\phi\over 2}}{\rm
and}\ |a|^{2}+|b|^{2}=1$ (18)

with superposition coefficients $a$ and $b$.

We now know the quantum probabilities, namely $\cos^{2}{\theta\over 2}$ and
$\sin^{2}{\theta\over 2}$, for the state $|\theta,\phi\rangle$ to collapse as
a result of a measurement to the state $|0,0\rangle$ or the North Pole of the
Bloch sphere, or the state $|0,\pi\rangle$ or the South Pole of the Bloch
sphere, respectively. That information gives us a way to determine the
mathematical form in the two-dimensional complex Hilbert space that the state
$|\theta,\phi\rangle$ possesses. Indeed, lets write the most general
mathematical form for this state $|\theta,\phi\rangle$, namely

$\displaystyle|\theta,\phi\rangle=a|0,0\rangle+b|\pi,0\rangle$ (19)

where $a$ and $b$ are two complex numbers such that $|a|^{2}+|b|^{2}=1$, we
just apply the superposition principle here, one of the basic principles of
quantum theory. And then we must have

$\displaystyle|a|^{2}=\cos^{2}{\theta\over 2}\quad{\rm
and}\quad|b|^{2}=\sin^{2}{\theta\over 2}$ (20)

We can solve these equations and should not forget that $a$ and $b$ are
complex numbers. A straightforward solution is $a=\cos{\theta\over 2}$ and
$b=\sin{\theta\over 2}$, but that is not the general solution. Note that a
dependence of a solution on $\phi$ must vanish in $|a|^{2}$ and $|b|^{2}$,
which means that this dependence on $\phi$ is in the form of a phase factor
$e^{i\phi}$, since such a phase factor disappears if the absolute value of the
complex numbers is calculated. This is how we arrive at the general solution
$a=\cos{\theta\over 2}e^{-i{\phi\over 2}}$ and $b=\sin{\theta\over
2}e^{i{\phi\over 2}}$, which we introduced in (23), noting that without loss
of generality we can multiply with a phase factor such that this specific form
arises 444Another choice often found in textbooks is $a=\cos{\theta\over 2}$
and $b=\sin{\theta\over 2}e^{i\phi}$, which equals the solution we use in this
article except a phase of $e^{-i{\phi\over 2}}$.. It can be shown that the
complex numbers as a number system on which the vector space is built are
essential, it is not possible to obtain the probability structure of a qubit
with a two-dimensional vector space over the real numbers (Aerts, 1986).

What we are particularly interested in for our present article is the fine
structure of the mechanism in the measurement in the extended Bloch model
shown in Figures 1 and 2. And before we continue, we wish to note the
following. The example depicts a two-dimensional quantum entity and a
measurement with two final states $A_{up}$ and $A_{down}$, hence two outcomes.
However, in later years, it was shown that a similar model can be built for a
$n$-dimensional quantum entity and a measurement with an arbitrary number $n$
of final states, and thus an arbitrary number $n$ of outcomes (Aerts & Sassoli
de Bianchi, 2014). We will not explicitly describe these higher dimensional
extended Bloch models in the present article, but mention that the simple
geometric properties of the configurations lead to exactly the quantum
probabilities in a completely similar way than this is the case for this two
dimensional quantum entity. The details of the analysis that we will now make
for the two dimensional quantum entity can also be made for the higher
dimensional quantum entities in a similar way. Of course, that we find exact
quantum probabilities in these measurement models also depends on a certain
symmetry that we introduced at the measurement level, namely that the elastic
breaks with a randomness that is ‘uniformly’ distributed. If the elastic does
not possess this symmetry property, we will still find probabilities that do
not fit a classical probability model, but they will not be perfect quantum
probabilities, e.g. a complex Hilbert space model will not be able to
represent them, they will thus be rather quantum-like. More general quantum
formalism than this of standard quantum mechanics in a complex Hilbert space,
and not based on vectors representing states, can then be used (Aerts, 1982,
1992; Gudder & Zanghi, 1984).

To verify that quantum measurement gives rise to the mechanism of categorical
perception, we must be able to express for states how much they differ from
each other. But before we get to that, we must identify which states play the
role of stimuli, and which states play the role of percepts in a quantum
measurement process. The stimuli are represented by the states of the quantum
entity independent of any measurement that would be going on, thus they are
what in the quantum jargon are called the pure states. In Hilbert space, these
pure states are represented by unit vectors of the Hilbert space. In the Bloch
representation, it is the points of the Bloch sphere that represent the pure
states of the considered quantum entity.

In what happens during a measurement as we represent it in the extended Bloch
model, not only the places corresponding to points on the surface of the Bloch
sphere play a role, but also places corresponding to points of the interior of
the Bloch sphere are important, for example the point $A^{\prime}$ where the
ball lands and sticks to the elastic after falling orthogonally on it. The
points over which the elastic is stretched are also in the interior of the
Bloch sphere. Besides the vector space formalism of quantum mechanics, where
the main role is played by the complex Hilbert space, which is a vector space
over the complex numbers, there also exists a density matrix formalism of
quantum mechanics. However, both formalism have a fundamentally different
mathematical structure, the Hilbert space is a vector space, as we already
mentioned, while the set of density matrices is a convex space. Convex
combinations of density matrices give another density matrix, while linear
combinations of vectors give another vector. Also, both structures can live
together on an underlying mathematical entity, e.g. density matrices are found
in the Hilbert space formalism as density operators. Both are also found in
the Bloch representation, the unit vectors of Hilbert space as the points of
the surface of the Bloch sphere, and the density matrices as the points of the
interior of the Bloch sphere. Let us introduce more systematically these
mathematical structures by writing the different vectors with respect to the
canonical basis.

$\displaystyle|1,0\rangle=(1,0)$ (21) $\displaystyle|0,1\rangle=(0,1)$ (22)

We have

$\displaystyle|\theta,\phi\rangle=(\cos{\theta\over 2}e^{-i{\phi\over
2}},\sin{\theta\over 2}e^{i{\phi\over 2}})$ (23)

We already mentioned the density matrix formalism as also a quantum mechanical
formalism. The points of the interior of the Bloch sphere correspond to such
density matrices of this quantum formalism. We will denote a density operator
corresponding to the point with spherical coordinates $(r,\theta,\phi)$,
$\rho\in[0,1]$, $\theta\in[0,\pi]$, $\phi\in[0,2\pi]$ as
$D_{(r,\theta,\phi)}$. Let us use some of the known properties to calculate
the density matrix for some of the points of the Bloch sphere and of its
interior. We already introduced the density matrix corresponding to a pure
state in (12), hence in the notation we introduced this is
$D_{(1,\theta,\phi)}$. Let us calculate the density matrices of the North and
South Poles of the Bloch sphere, or of the points $A_{up}$ and $A_{down}$. We
have

$\displaystyle D_{(1,0,\phi)}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\cos^{2}{\theta\over 2}&\cos{\theta\over
2}\sin{\theta\over 2}e^{-i\phi}\\\ \cos{\theta\over 2}\sin{\theta\over
2}e^{i\phi}&\sin^{2}{\theta\over
2}\end{pmatrix}_{\theta=0,\phi}=\begin{pmatrix}1&0\\\ 0&0\end{pmatrix}$ (24)
$\displaystyle D_{(1,\pi,\phi)}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\cos^{2}{\theta\over 2}&\cos{\theta\over
2}\sin{\theta\over 2}e^{-i\phi}\\\ \cos{\theta\over 2}\sin{\theta\over
2}e^{i\phi}&\sin^{2}{\theta\over
2}\end{pmatrix}_{\theta=0,\phi}=\begin{pmatrix}0&0\\\ 0&1\end{pmatrix}$ (25)

Let us now calculate the density operator $D_{A^{\prime}}$ representing the
state of the entity when it is in the point $A^{\prime}$ as in Figure 2. We
remark that $A^{\prime}$ lies on the line between $A_{down}$ and $A_{up}$
sticking on the elastic which is stretched between $A_{down}$ and $A_{up}$ on
this line. Making use of a general property of the set of all density
operators, i.e. that it is a set closed by convex combination, we know that
$D_{A^{\prime}}$ is a convex combination of $D_{(1,\pi,\phi)}$ and
$D_{(1,0,\phi)}$, which gives

$\displaystyle D_{A^{\prime}}=\lambda\begin{pmatrix}0&0\\\
0&1\end{pmatrix}+(1-\lambda)\begin{pmatrix}1&0\\\
0&0\end{pmatrix}=\begin{pmatrix}1-\lambda&0\\\ 0&\lambda\end{pmatrix}$ (26)

for $\lambda\in[0,1]$. From Figure 2, given that $A^{\prime}$ is obtained by
projecting orthogonally to the line between $A_{down}$ and $A_{up}$, we have
that $A^{\prime}$ lies on the line between $A$ and the point with spherical
coordinates $(1,\theta,\phi+\pi)$, to which corresponds the density operator

$\displaystyle D_{(1,\theta,\phi+\pi)}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\cos^{2}{\theta\over 2}&\cos{\theta\over
2}\sin{\theta\over 2}e^{-i(\phi+\pi)}\\\ \cos{\theta\over 2}\sin{\theta\over
2}e^{i(\phi+\pi)}&\sin^{2}{\theta\over 2}\end{pmatrix}$ (27) $\displaystyle=$
$\displaystyle\begin{pmatrix}\cos^{2}{\theta\over 2}&-\cos{\theta\over
2}\sin{\theta\over 2}e^{-i\phi}\\\ -\cos{\theta\over 2}\sin{\theta\over
2}e^{i\phi}&\sin^{2}{\theta\over 2}\end{pmatrix}$

This means that we have

$\displaystyle D_{A^{\prime}}=\mu
D_{(1,\theta,\phi)}+(1-\mu)D_{(1,\theta,\phi+\pi)}$ (28)

for $\mu\in[0,1]$. From (26) and (28) follows that we must have

$\displaystyle\mu\cos{\theta\over 2}\sin{\theta\over
2}e^{-i\phi}-(1-\mu)\cos{\theta\over 2}\sin{\theta\over 2}e^{-i\phi}=0$
$\displaystyle\Leftrightarrow\mu-(1-\mu)=0$
$\displaystyle\Leftrightarrow\mu={1\over 2}$ (29)

and

$\displaystyle\lambda=\sin^{2}{\theta\over 2}$ (30)

This gives us

$\displaystyle D_{A^{\prime}}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\cos^{2}{\theta\over 2}&0\\\
0&\sin^{2}{\theta\over 2}\end{pmatrix}=\cos^{2}{\theta\over
2}\begin{pmatrix}1&0\\\ 0&0\end{pmatrix}+\sin^{2}{\theta\over
2}\begin{pmatrix}0&0\\\ 0&1\end{pmatrix}$ (31) $\displaystyle=$
$\displaystyle\cos^{2}{\theta\over 2}D_{A_{up}}+\sin^{2}{\theta\over
2}D_{A_{down}}$

Let us note that it follows from (28) that we also know the density matrix of
each point lying on the line between the two points of the surface of the
Bloch sphere, the point $A$ with coordinates $(1,\theta,\phi)$ and the point
with coordinates $(1,\theta,\phi+\pi)$, which is mirrored with respect to the
North-South axis of the Bloch sphere. For a value of $\mu$ ranging from -1 to
+1, the density matrix is given by

$\displaystyle D_{A^{\prime},\mu}$ $\displaystyle=$ $\displaystyle\mu
D_{(1,\theta,\phi)}+(1-\mu)D_{(1,\theta,\phi+\pi)}$ (32) $\displaystyle=$
$\displaystyle\begin{pmatrix}\cos^{2}{\theta\over 2}&\mu\cos{\theta\over
2}\sin{\theta\over 2}e^{-i(\phi+\pi)}\\\ \mu\cos{\theta\over
2}\sin{\theta\over 2}e^{i(\phi+\pi)}&\sin^{2}{\theta\over 2}\end{pmatrix}$

the density matrix of the point corresponding to the value of $\mu$ on the
line between the points $(1,\theta,\phi+\pi)$ and $(1,\theta,\phi)$. The
parameter $\mu$ occurs only in the diagonal terms of the density matrix, and
the orthogonal falling of the ball on the elastic takes place on the line
connecting the two points $(1,\theta,\phi+\pi)$ and $(1,\theta,\phi)$, and
that orthogonal falling occurs when $\mu$ goes from 1 to 0. In the jargon of
quantum mechanics, the decreasing of the non-diagonal terms of the density
matrix is named ‘decoherence,’ and is described as a phenomenon of decreasing
quantum nature and increasing classical nature.

Let us briefly summarize the elements of a quantum measurement and its Bloch
sphere representation. Each point $(r,\theta,\phi)$, $r\in[0,1]$,
$\theta\in[0,\pi]$, $\phi\in[0,2\pi]$, within the sphere shown in Figure 2,
corresponds to a density operator $D_{(r,\theta,\phi)}$, and the points of the
sphere surface, i.e. $r=1$, correspond to pure states of the quantum entity.
The points of the interior of the sphere correspond to density states of the
quantum entity. With a measurement corresponds a convex subspace of the
interior of the sphere, in the case of the measurement with possible final
states after the measurement, $A_{up}$ and $A_{down}$, this convex subspace is
given by the line segment from $(1,\pi,\phi)$ to $(1,0,\phi)$, where the
elastic is stretched. But all lines passing through the center of the sphere
in Figure 2 harbor such an equivalent measurement. The points on the line all
correspond to a density state, except for the end points, which are indeed
part of the surface of the sphere, and thus correspond to a pure state. When a
measurement commences, on an entity that is in a pure state, such as this one
in the point $A$, an expectation is allowed to play a role of what this state
in the point $A$ means to the measuring apparatus located on a line between
two diametrical points of the sphere surface, such as the points $A_{down}$
and $A_{up}$. That is the meaning of the orthogonal projection which brings
the state in $A$, to a state in $A^{\prime}$. Once the state has become the
one corresponding to the point $A^{\prime}$, then the entity is in a density
state. The transformation from the state corresponding to $A$ to the state
corresponding to $A^{\prime}$ can be read on the matrix representation, namely
it is the non-diagonal terms of the matrix of the density state that disappear
to arrive at the state corresponding to $A^{\prime}$.

##  4 Categorical Perception and Quantum Measurement

As we already mentioned, in this article we want to show that a quantum
measurement structurally incorporates the mechanism of categorical perception.
At the same time, we want to propose an interpretation of ‘what exactly could
be imagined to happen during a quantum measurement’ with the intention of
representing the ‘cognition aspect’ of this event as clearly as possible. In
this sense, our work also provides additional evidence for the ‘conceptuality
interpretation’ of quantum mechanics an original interpretation (Aerts, 2009b)
that is being further developed in our Brussels research group (Aerts et al.,
2020). Indeed, the basic hypothesis of the conceptuality interpretation is
that a quantum measurement has the characteristics of a cognitive process, as
a consequence the quantum entities which are the subject of the measurement
have a conceptual nature. When we mention that the basic hypothesis of
conceptuality interpretation consists in postulating the presence of a
cognitive interaction in the micro world we do not mean that this is a
cognitive interaction very similar to the human cognitive interaction, perhaps
and even probably so, it is of an alien nature. We do mean that there are some
basic aspects present that we also find in the human cognitive interaction,
for example ‘the use of concepts as communication tools’ (Aerts, 2009b; Aerts
et al., 2020). If the phenomenon of categorical perception can be shown to be
present in what is occurring during a quantum measurement then this brings
extra evidence for the hypothesis that a quantum measurement is a cognitive
process. Ultimately, we also want to show that it makes sense to consider
colors as the quanta of light for human visual perception similar to how
photons are the quanta of light for a physical measurement device during a
quantum measurement.

To inspire our interpretation of what could be taking place during a quantum
measurement, we present an experiment conducted by Jerome Bruner and Postman,
in which elements of human perception are brought out (Bruner & Postman,
1949). Note that Bruner, considered to be one of the important psychologists
of the twentieth century, and Postman worked on perception in the 1940’s, an
era where cognitive science was not yet defined as a sub discipline in
psychology. In the experiment we will describe in some detail, Bruner and
Postman’s intention was to confront participants with stimuli they called
‘incongruent’. To make clear what this term ‘incongruent’ means, we outline
the view on perception in which Bruner and Postman framed the experiment. They
worked within a view of ‘perception’ in which the hypothesis is advanced that
a perception corresponding to a stimulus is an event in which there is, what
they call, a ‘construction-defense’ balance at work. From a pattern of
expectations, a structure present in the mind of the one who perceives,
‘construction’ takes place when a stimulus meets an expectation, while
‘defense’ takes place when a stimulus counters an expectation. In the
experiment we will now describe, participants are confronted with stimuli that
confirm but also contradict the structure of expectations with which
participants’ minds take part in the experiment. The purpose of the experiment
is to distinguish and analyze different phases of dealing with this
contradiction, which Bruner and Postman name ‘incongruity’. We are
particularly interested in one of the phases, which they call the ‘dominance
reaction’. Let us now first describe the experiment in Bruner & Postman (1949)
and then further on specify which aspects we will use to craft our
interpretation of a quantum measurement.

Twenty-eight participants, students at Harvard and Radcliffe, were shown
successively by tachistoscopic exposure five different playing cards. From one
to four of these cards were incongruous – color and suit were reversed. Order
of presentation of normal and incongruous cards was randomized. The normal and
‘trick’ cards used were the following. Normal cards (printed in their proper
color) five of hearts, ace of hearts, five of spades, seven of spades. Trick
cards (printed with color reversed), three of hearts (black), four of hearts
(black), two of spades (red), six of spades (red), ace of diamonds (black),
six of clubs (red). Fourteen orders of presentation were worked out, and two
subjects were presented the cards in each of these orders. There were three
types of stimulus series (1) a single trick card embedded in a series of four
normal cards, (2) a single normal card embedded in a series of four trick
cards, (3) mixed series in which trick and normal cards were in the ratio of 3
to 2 or 2 to 3. Each card was presented successively until correct recognition
occurred, three times each at 10 ms, 30 ms, 50 ms, 70 ms, 100 ms, 150 ms, 200
ms, 250 ms, 300 ms, 400 ms, 450 ms, 500 ms, and then in steps of 100 ms to
1000 ms. If at 1000 ms recognition did not occur, the next card was presented.
In determining thresholds, correct recognition was defined as two successive
correct responses. At each exposure, the subject was asked to report
everything he or she saw or thought he or she saw. The cards were mounted on a
medium gray cardboard and were shown in a Dodge-Gerbrands tachistoscope. The
pre-exposure field was of the same gray color and consistency as the exposure
field save that it contained no playing card. The light in the tachistoscope
was provided by two G E. daylight fluorescent tubes.

For those readers who would like more details about the experiment and its
analysis and conclusions, we refer to (Bruner & Postman, 1949), we are
particularly interested in the different reactions of the participants, and
especially this one that Bruner and Postman called the ‘dominant reaction’.
This dominant reaction to the cards shown in swift consists of providing a
response that is entirely within the expectation pattern of the participant,
namely that these are standard cards that have not been tricked. With respect
to the tricked cards, this reaction consists, essentially, of a ‘perceptual
denial’ of the incongruous elements in the stimulus pattern. Faced with a red
six of spades, for example, a subject may report with considerable assurance,
‘the six of spades’ or the ‘six of hearts’, depending upon whether he is color
or form bound. In the one case the form dominates and the color is assimilated
to it, in the other the stimulus color dominates and form is assimilated to
it. In both instances the perceptual resultant conforms with past expectations
about the ‘normal’ nature of playing cards. We will not describe the other
possible reactions further, as we are concerned with this one, incidentally,
most common reaction. With longer times of showing the cards, and repeating
the experiment with the same participants, a different pattern does develop,
with the participants eventually finding out that tricked cards are being used
– a real repetition of the experiment, to calculate e.g. the collapse
probabilities, should, for this reason, always invite other participants.

In Bruner and Postman’s work, perception is considered a phenomenon that, in
addition to the influence of the stimulus, is also determined by the presence
of the expectation pattern of the one who perceives. True, the model of human
perception proposed by them contains elements that are probably only relevant
to the human specificity of what perception and cognition are. And, if as a
basic hypothesis of the conceptuality interpretation of quantum mechanics the
quantum measurement is considered as a cognitive process, in which then also a
perceptual process takes place, then, like we mentioned already, it is not
meant that the specific properties of human cognition and perception will also
be present. In this sense, we are primarily interested in the fundamental and
general properties of cognitive and perceptual processes, the formation of
concepts itself probably being the most important one. The evidence gathered
so far in support of the conceptuality interpretation is largely based on the
hypothesis that quantum entities are conceptual entities, hence that the
cognitive process makes use of ‘concepts’. The structure underpinning the
phenomenon of categorical perception is already more specific, but still
fundamental and general enough so that, given its abundance in human
cognition, it makes sense to investigate its presence in a quantum
measurement. Its identification in a quantum measurement will also allow us,
as we will show, to distinguish precisely its most fundamental and general
properties. But first, we want to point out some of these properties inspired
by Bruner and Postman’s view of human perception.

According to Bruner and Postman, the set of percepts is primarily a set of
‘expectations related to the stimuli’. We will, for a quantum measurement,
identify this set of percepts with the set of points on the centerline of the
Bloch sphere, that is, where in our extended Bloch model the elastic is
stretched. Where are the stimuli for a quantum measurement? The stimuli are
assumed to be independent of any measurement and so, for a quantum
measurement, they are the pure states of the entity in question, and within
the Bloch sphere representation they are located in the points of the surface
of the sphere. The orthogonal falling of the ball of a point of the sphere
surface on the elastic band represents the interaction of the cognitive
apparatus with the entity under consideration. This means that the
transformation from a point on the surface of the Bloch sphere, say the point
$A$, to a point on the centerline where the elastic is stretched, say the
point $A^{\prime}$, represents the interaction between expectation, starting
from the cognitive apparatus, on the one hand, and perception, starting from
the considered entity, on the other.

If we want to give an expression to the mechanism of categorical perception,
we must be able to estimate and/or measure distances between both pure states,
the stimuli, located in points on the surface of the Bloch sphere, and density
states, the percepts, located in the points of a centerline of the Bloch
sphere. Using these distances, the differences between stimuli and the
differences between percepts can be expressed quantitatively. However, we must
be careful in choosing how we will measure distances in the Bloch sphere
representation, since there is no unique obvious metric on the whole set of
quantum states. One of the well-studied partial metrics on the set of quantum
states is the trace distance

$\displaystyle T(D_{1},D_{2})$ $\displaystyle=$ $\displaystyle{1\over
2}Tr\left[\sqrt{(D_{1}-D_{2})^{*}(D_{1}-D_{2})}\right]$ (33)

It can be shown that for a two-dimensional quantum entity, i.e., a qubit,
modeled with the Bloch sphere, the trace distance is equal to half the
Euclidean distance in the three-dimensional Euclidean space of which the Bloch
sphere is a part. For two pure states $|\psi_{1}\rangle$ and
$|\psi_{2}\rangle$, hence represented by two points of the surface of the
Bloch sphere, it can be shown that the trace distance is given by

$\displaystyle
T(|\psi_{1}\rangle\langle\psi_{1}|,|\psi_{2}\rangle\langle\psi_{2}|)$
$\displaystyle=$ $\displaystyle\sqrt{1-|\langle\psi_{1}|\psi_{2}\rangle|^{2}}$
(34)

That means, for example, that the trace distances between the quantum state
where $A$ is and the North and South Pole of the Bloch sphere are given by

$\displaystyle T(D_{(1,\theta,\phi)},D_{(1,0,0)})$ $\displaystyle=$
$\displaystyle\sqrt{1-|\langle\theta,\phi|0,0\rangle|^{2}}=\sqrt{1-|\langle(\cos{\theta\over
2}e^{-i{\phi\over 2}},\sin{\theta\over 2}e^{i{\phi\over
2}})|(1,0)\rangle|^{2}}$ (35) $\displaystyle=$
$\displaystyle\sqrt{1-\cos^{2}{\theta\over 2}}=\sin{\theta\over 2}$
$\displaystyle T(D_{(1,\theta,\phi)},D_{(1,\pi,0)})$ $\displaystyle=$
$\displaystyle\sqrt{1-|\langle\theta,\phi|\pi,0\rangle|^{2}}=\sqrt{1-|\langle(\cos{\theta\over
2}e^{-i{\phi\over 2}},\sin{\theta\over 2}e^{i{\phi\over
2}})|(0,1)\rangle|^{2}}$ (36) $\displaystyle=$
$\displaystyle\sqrt{1-\sin^{2}{\theta\over 2}}=\cos{\theta\over 2}$

We can now see interesting connections using the simple geometry of the Bloch
representation, let us consider Figure 3. It represents the Bloch
representation with $\phi={\pi\over 2}$, hence $A$ laying in the $zy$-plane.
The triangle $A_{up}$, $A$, $A_{down}$ is a right-angled triangle with right
angle in $A$. The hypothenuse of this right-angled triangle is the centerline
connecting $A_{up}$ to $A_{down}$ and it has length equal to $2$.

The angle in $A_{down}$ of the triangle is equal to half the angle $\theta$,
and so the length of the line segment connecting $A_{up}$ to $A$ is equal to
$2\sin{\theta\over 2}$. In a similar manner, the figure shows us that the
length of the line segment connecting $A$ and $A_{down}$ is given by
$2\cos{\theta\over 2}$. This agrees with the calculation of the trace distance
we made for the states connected to corresponding points of the Bloch sphere,
as given in (35) and (36), reminding us that for a qubit the trace distance is
equal to half the Euclidean distance. This reasoning and additional
calculations, also the general expression for the trace distance between pure
states, makes it clear that the trace distance is not the metric to be used
between pure states. Indeed, any effect due to quantum coherence disappears
when using this metric, since in the general expression (34) the only
reference to the states is the inner product between the two states, and hence
traces of quantum coherence effects are no longer present.

There is also a purely geometric way by which we can see that the trace
distance is unsuitable for measuring distances between pure states. Consider
any two pure states, which gives us two points of the surface of the Bloch
sphere. If we connect these two points with a line, we see that the points of
the line different from these two points always lie in the interior of the
Bloch sphere and thus correspond to density states. It is clear that defining
a distance associated with a specific geometric entity, the straight line in
the three-dimensional Euclidean space to which the Bloch sphere belongs,
carries limitations when, as is the case with pure states, this distance is
intrinsically intended for elements of a subset with a structure in which the
straight line is not contained, and we mean here the spherical surface of the
Bloch sphere, to which the pure states are restricted.

Figure 3: Using the simple geometry of the Bloch sphere, we can recover the
trace distance we calculated in (35) and (36) as illustrated in this figure.
The triangle $A_{up}$, $A$, $A_{down}$ is rectangular which makes its angle in
$A_{down}$ equal to ${\theta\over 2}$, and consequently, knowing that the
distance between $A_{up}$ and $A_{down}$ is equal to $2$, the figure shows us
that the distance between $A_{up}$ and $A$ is equal to $2\sin{\theta\over 2}$,
and the distance between $A_{down}$ and $A$ is equal to $2\cos{\theta\over
2}$. Since the trace distance is equal to half the Euclidean distance for the
Bloch sphere of a qubit, we have derived the results of (35) and (36) from the
geometry of the Bloch sphere.

So we must ask ourselves at this point of our analysis which the proper metric
to be used on pure quantum states is. A natural notion of distance between
pure states should only account for the pure states, that are in-between the
points on the Bloch sphere representing these states, the length of the
circular arc connecting these points would be such a quantity. More precisely,
if $\theta$ and $\alpha$ are the polar angles of the pure states
$|\theta,\phi\rangle$ and $|\alpha,\phi\rangle$ (see Figure 4), the distance
to be used to measure how much they are separated from each other, normalized
to 1, would then be

$\displaystyle
d_{pure}(|\theta,\phi\rangle,|\alpha,\phi\rangle)={1\over\pi}|\theta-\alpha|$
(37)

Note that it is sufficient to define the distance for two pure states with
equal azimuthal angle $\phi$. Indeed, we can always choose a North Pole and
South Pole for any two pure states such that both states lie in the plane with
equal azimuthal angle, so that then the arc of the circle through both points
is determined by the difference of the polar angles of both states.

This distance also results from the angle between pure states calculated from
their Hilbert space inner product, which shows that it is a natural distance
associated with the Bloch representation of pure states. The notion of
fidelity

$\displaystyle
F(D_{1},D_{2})=\left(tr\sqrt{\sqrt{D_{1}}D_{2}\sqrt{D_{1}}}\right)^{2}$ (38)

which can also be defined on the density states, generalizes this inner
product, because for pure states $\psi_{1}$ and $\psi_{2}$ we have

$\displaystyle F(\psi_{1},\psi_{2})=|\langle\psi_{1}|\psi_{2}\rangle|^{2}$
(39)

but does not give rise to a metric over the entire set of quantum states. But
if we consider angles, or equivalently circular arcs, as a measure of the
natural distance between pure states in the Bloch model, we are actually using
the notion of fidelity to evaluate differences between stimuli.

So, we will use two different notions of distance, one that considers the
circular arc between two vector-states, at the surface of the Bloch sphere,
and the other one which considers the Euclidean distance between density
states, inside the Bloch sphere. The normalized to 1 distance between two such
decohered density states

$\displaystyle\begin{pmatrix}\cos^{2}{\theta\over 2}&0\\\
0&\sin^{2}{\theta\over 2}\end{pmatrix}\quad{\rm
and}\quad\begin{pmatrix}\cos^{2}{\alpha\over 2}&0\\\ 0&\sin^{2}{\alpha\over
2}\end{pmatrix}$ (40)

will be given by (see Figure 4)

$\displaystyle d_{density}(\theta,\alpha)={1\over 2}|\cos\theta-\cos\alpha|$
(41)

Equipped with these two distances, let us now analyse how the phenomenon of
categorical perception is naturally expressed in a quantum measurement
process. As we mentioned, the points of the sphere’s diameter where the
elastic band is stretched, is where the percepts lie, for the measurement in
question. These percepts represent an ‘expected reality’, which however is not
to be understood as a collection of wild speculations, but rather as the best
picture of what is real in relation to the meaning carried by the measured
entity, but adapted to the context which the region where the elastic is
stretched represents within the mathematical model of the quantum measurement.
So the region where the elastic lies does not simply represent an ‘expected
reality’, as might be misunderstood from perception theory a la Bruner and
Postman, and from the experiment we described, a more correct way to describe
what this region represents is the following. ‘That which can be put forward
from the context of the measuring apparatus and the cognitive process that
takes place as the best hypothesis about reality’. And, as a form of
contextual limitation, portions of expected reality creep in to this as a
consequence of attempting the best hypothesis about non-contextual reality.
Also some contextual biases cannot be avoided, and the warping of categorical
perception is one of them. More precisely still, the one who perceives is
actively engaged during a measurement process and his or her ‘expected
reality’ plays a role in what will occur, together with the reality of the
measured entity, expressed by its pure state, which is instead measurement
independent. At the level of the contextual region where the elastic lies, the
model is Kolmogorovian, the probabilities being an expression of the lack of
knowledge of the one who perceives. This is realized in the extended Bloch
model by the unpredictable point $\lambda$ where the elastic breaks. Also, the
connection between the reality of the considered entity (the pure states
describing the stimuli) and the elements of the contextual reality (the on-
elastic density states describing the percepts for the given measurement), is
illustrated by the deterministic orthogonal fall of the point particle
representative of the stimulus onto the elastic band, transforming it into a
percept, thus reaching a stage where an outcome (an answer) is actualized, and
we are back to a pure state.

To see how a quantum measurement brings about the warping effect of
categorical perception, let us consider, to fix ideas, a situation where only
two colors exist, Light and Dark, so we are precisely in a situation that can
be described in a three-dimensional Bloch sphere. Note that Eleanor Rosch
formulated the rationale for the prototype theory for concepts while teaching
colors to a primitive community in Papua New Guinea, whose language, called
Berinomo, has just two names for colors (Rosch 1973).

Let us locate the first color, Light, at the North Pole of the Bloch sphere,
and the second color, Dark, at its South Pole. At the equator, the transition
from Light to Dark will then occur (see Figure 4).

Figure 4: We consider a situation where there are two names for colors, which
we call Light and Dark, and wish to show that the quantum measurement model,
which in this case represents a ‘qubit’, incorporates the phenomenon of
categorical perception. For this, we consider three pure states
$|\pi/3,\phi\rangle$, $|2\pi/3,\phi\rangle$ and $|0,\phi\rangle$,
respectively, in the Bloch representation located in points $A$, $B$ and $C$,
which represent stimuli associated with a quantum measurement on this qubit.
With each of the three pure states corresponds a density state in which the
qubit is located after the measurement, localized in the Bloch representation
in respectively points $A^{\prime}$, $B^{\prime}$ and $C^{\prime}$, and
described by respectively density matrices (42), (43) and (44). The pure
states in $A$ and $B$ belong to two different colors, Light and Dark, and lie
at a distance 1/3 from each other. The density states corresponding to them,
located in points $A^{\prime}$ and $B^{\prime}$, are at a distance 1/2 from
each other. We see here the dilation mechanism of categorical reception at
work, for percepts belonging to different categories, Light and Dark. The pure
states in $C$ and $A$ belong to the same color, Light, and also lie at a
distance 1/3 from each other. The density states corresponding to $C$ and $A$,
located in points $C^{\prime}$ and $A^{\prime}$, are at a distance 1/4 from
each other. We see here the contraction mechanism of categorical reception at
work, for percepts belonging to the same category.

Let us then introduce three different pure states, the first one located in
point $A$ (see Figure 4) represented by the vector $|\pi/3,\phi\rangle$, hence
with polar angle $\theta=\pi/3$, the second one located in point $B$ (see
Figure 4) represented by the vector $|2\pi/3,\phi\rangle$, hence with polar
angle $\theta=2\pi/3$, and the third one located in the North Pole (see Figure
4), represented by the vector $|0,\phi\rangle$, (hence, this is the eigenstate
describing Light, with a polar angle $0$), assuming for simplicity that they
all lie on a same plane, hence have the same asimuthal angle $\phi$. When a
Light-Dark color-measurement is performed, the pre-measurement pure states
deterministically transform into the fully decohered pre-collapse density
states, obtained by plunging the associated point particles into the sphere,
orthogonally with respect to the line subtended by the Light and Dark outcome
states, which is the region of the percepts, i.e., of the ‘contextual reality’
relative to this specific color-measurement. This results for the pure states
located in $A$, $B$ and $C$, respectively, in density states located in points
$A^{\prime}$, $B^{\prime}$ and $C^{\prime}$, and given, respectively, by the
density matrices

$\displaystyle\begin{pmatrix}1/4&0\\\ 0&3/4\end{pmatrix},$ (42)
$\displaystyle\begin{pmatrix}3/4&0\\\ 0&1/4\end{pmatrix}$ (43)
$\displaystyle{\rm and}\quad\begin{pmatrix}1&0\\\ 0&0\end{pmatrix}$ (44)

If we consider the pure states $|{\pi\over 3},\phi\rangle$ and $|{2\pi\over
3},\phi\rangle$, located respectively in points $A$ and $B$, they are in the
Light and Dark hemispheres of the Bloch sphere, hence they are two different
colors, and we can now easily see that their transformation to the
corresponding decohered density states, located respectively in points
$A^{\prime}$ and $B^{\prime}$ (see Figure 4), and described respectively by
the density matrices (42) and (43) exhibits a warping that is a dilation.
Indeed, their distance is $1/3$, i.e., one third of the maximal distance
between two stimuli, while the associated percepts, corresponding to the
decohered quantum states, have a distance of $1/2$, i.e., one half of the
maximal distance between two percepts. On the other hand, if we consider the
pure states $|0,\phi\rangle$ and $|{\pi\over 3},\phi\rangle$, located
respectively in points $C$ and $A$, belonging to the same color, namely the
color Light, the opposite warping occurs. Indeed, on the stimuli side, the
distance is again $1/3$, whereas on the percepts side, the corresponding
density states being located in points $C^{\prime}$ and $A^{\prime}$,
respectiveky, and described by the density matrices (44) and (42), we now have
distance $1/4$. This means that a warping takes place which is now a
contraction. This shows how the phenomenon of categorical perception is built
into a the quantum measurement.

##  5 Visual Perception, Colors and Quanta

Let us analyze what the events are that correspond to this way of using the
quantum measurement model to model the phenomenon of categorical perception of
colors. And let us start from the situation where the stimulus, as measured by
a physical measuring device that measures the frequency of electromagnetic
radiation, is perceived by a human being, and from what we know takes place
then.

Suppose the light being looked at has a frequency of 595 Terahertz, given that
1 Terahertz equals $10^{12}$ Hertz, which corresponds to a wavelength of about
504 nanometers. If we consult one of the many color spectra of visible light
available on the World-Wide Web, we can see that 504 nanometers wavelength
corresponds to a color right in between green and blue, hence light of this
frequency possesses a Blue-Green color, but it is not obvious whether when
someone is asked to classify this light among one of the basic or prototype
colors, whether that person will pick either Blue or Green. Each region that
makes a transition from one basic color to another basic color contains a
strip where it is really not clear which of the adjacent basic colors will be
chosen if such a choice is imposed, for example in an experiment.

That being said, we plan an experiment where we will show certain shades of
color and on a Likert scale have participants choose to classify these shades
of color with one of the basic colors. We will then investigate the extent to
which we can model the shades of color in a complex Hilbert space as
superpositions of the basis vectors each representing one of the basic colors.
Let us note that in our example of the shade Blue-Green carried by light of
frequency 595 Terahertz, this is a color carried by very pure light of exact
frequency. We already noted that the prototype colors for the English language
are ‘not’ the colors of the rainbow, for example the color Brown is not
realized by light of a specific frequency. Colors of reflected light can
represent very complex mixtures of different frequencies, just think of how
white light is obtained by rotating a color disk rapidly, what happens there
is very complex when the stimulus in question, the light shone from the
rotating disk, is examined from physics. For what colors ‘are’, what happens
in our eye on the retina, but especially in our brain, and perhaps just as
importantly, in and with our mind, is fundamental. Given the eleven prototype
colors considered as basic colors by Berlin & Kay (1969) and Rosch (1973), for
our planned experiment we will also consider these colors as basic colors.
This means that the set of all superpositions are the elements, more precisely
the unit vectors, of an eleven-dimensional complex Hilbert space, with a basis
of orthonormal vectors each of which represents one of the considered basic
colors. Although we can also explicitly build the Bloch representation for
this situation of eleven basic colors, and we plan to do so after we have
collected the data from our experiment, we will not become explicit about it
in this article, because the more than two-dimensional Bloch representations
are more complicated, and we have focused, at least in part also for reasons
of simplicity and intelligibility, but also because we wanted to turn to the
fundamental in this article, to the situation of two colors, Light and Dark.
After all, the fundamental, namely the way in which stimuli and expectations
play a role in determining percepts, is equally and fully present in this
situation of two colors.

So let us return to the situation of two colors that we have called Light and
Dark. Let it also be noted here that a restriction of vision to Light and
Dark, and thus shades of gray, still leads to a powerful use of the sense of
‘sight’. Indeed, there are several animals that see only shades of gray, and
this group is called the ‘monochromatic mammals’. Several bats, rodents and
the common raccoon are among them. One common denominator among these
mentioned species is that they are nocturnal animals, so their monochromatic
vision gives them a special advantage at night. The shade of gray that
contains the most uncertainty regarding its classability as Light or Dark is
the shade that lies at the equator of the Bloch sphere (see Figure 4). It is
plausible to associate a probability equal to 1/2 with the point at the center
of the elastic that runs between North and South Poles in the extended Bloch
model.

Are the probabilities proposed by the Bloch model also obtained in the limit
of the fractions obtained for the outcomes when an experiment is conducted to
measure them? Regarding this question, we should mention the result of
research published in Aerts & Sassoli de Bianchi (2015). If we imagine
different individuals and the way they will decide to classify a particular
shade of gray as Light rather than Dark or vice versa, it is very plausible to
assume that there will be a fair amount of individual differences. Some will
still classify grays that are above the equator as Dark while others will
still classify grays that are below the equator as Light. It is shown in Aerts
& Sassoli de Bianchi (2015) that the average over such individual differences
tends to converge to a uniform distribution, which would then correspond to
the quantum probabilities proposed in the previous sections of this paper.
That is, the quantum measurement model would be a kind of first-order
approximation to more complex measurement situations, which would then also
explain why in cases where control over such individual differences is absent,
it leads to good predictive models. It is beyond the scope of our current
article to explain this complex situation of ‘quantum as first order
approximation’ in more detail, we refer the interested reader to Aerts &
Sassoli de Bianchi (2017) for a very detailed account of this situation.

Let us now come to our argument in connection with colors being quanta of
light for human vision, and so for a raccoon these are only two colors Light
and Dark, while for English-speaking humans they are eleven, White, Black,
Red, Yellow, Green, Blue, Brown, Purple, Pink, Orange and Gray. Let us explain
why these eleven colors are quanta of light for English-speaking humans.
Suppose a friends has a bicycle colored with exactly the color of light
corresponding to a frequency of 595 Terahertz. And suppose for a moment that a
conversation develops about this bicycle, perhaps it is lost and they are
looking for it. The color might then come up. Perhaps Green will be specified
as the color when asked, or possibly Blue will be named. Or, when precision is
important, Blue-Green is said. All three possibilities bring probative value
to our claim that the eleven prototype colors are quanta of light for English-
speaking people. Indeed, even if Blue-Green is given as the color for the
bicycle to be found, this word Blue-Green is a combination of two other words,
Blue and Green. Consider the pet-fish problem we mentioned in Section 4.
Indeed, in quantum cognition, such combinations of words are described by
superpositions of the vectors that describe the participles of these
combinations (Aerts & Gabora, 2005a, b; Aerts et al., 2012). The cultural fact
that it was decided in English to include these eleven designations as
separate words in the English language, but not to introduce a separate word
for the color that light of 595 Terahertz shines around, is what determines
their status as quanta. Indeed, it is the dynamics produced by the phenomenon
of categorical perception, the mechanism that causes certain stimuli, that is,
in this case of colors, certain frequencies of light, to contract so that it
becomes meaningful within the cognitive interaction to classify them all as
Green, while other stimuli are pushed apart so that it becomes meaningful
within the same cognitive interaction to classify them as different colors,
Green and Blue, for example, that we wish to call ‘quantization’. It is our
hypothesis that the same contraction and pushing apart occurs in the quanta
that light shows to physical measuring devices and their cognitive activities,
so that, in the case of light, the photons emerge as quanta. The process of
quantization ends, or better stabilizes, as an optimum of this cognitive
interaction.

This means that as far as human cognition is concerned, the interaction that
takes place between the stimulus, on the one hand, and the expectation pattern
of the person perceiving, on the other hand, in order to arrive at perception,
is of a greater complexity than the experiment with the tricked cards
described in brunerpostman1949 suggests. It is likewise of greater complexity
than the dynamics of elastics breaking as we suggested in the extended Bloch
model. That does not mean, however, that these models are irrelevant to making
progress in the depth of what human perception and cognition are. Indeed, they
both contain essential elements of this human perception and cognition and
suggest that more complex interactions need to be expected. But, the result
demonstrated in Aerts & Sassoli de Bianchi (2015, 2017), makes it clear that
such a more complex situation of interaction between the measuring device and
the entity being measured should not necessarily lead to the need for a more
complex model of measurement, at least not if we are satisfied with the linear
model proposed by quantum mechanics being a first order approximation of what
takes place during measurement. So it is especially in comparison to a
classical description that assumes that no interaction at all takes place
between measuring device and the entity being measured that the first order
approximation of the linear quantum model is an essential step forward.

## References

  * Aerts (1982) Aerts, D. (1982). Description of many physical entities without the paradoxes encountered in quantum mechanics. Foundations of Physics 12, pp. 1131-1170. doi: 10.1007/BF00729621.
  * Aerts (1986) Aerts, D. (1986). A possible explanation of the probabilities of quantum mechanics. Journal of Mathematical Physics 27, pp. 202-210. doi: 10.1063/1.527362.
  * Aerts (1992) Aerts, D. (1992). The construction of reality and its influence on the understanding of quantum structures. International Journal of Theoretical Physics 31, pp. 1815-1837. doi: 10.1007/BF00678294.
  * Aerts (2009a) Aerts, D. (2009). Quantum structure in cognition. Journal of Mathematical Psychology 53, pp. 314-348. doi: 10.1016/j.jmp.2009.04.005.
  * Aerts (2009b) Aerts, D. (2009). Quantum particles as conceptual entities: A possible explanatory framework for quantum theory. Foundations of Science 14, pp. 361-411. doi: 10.1007/s10699-009-9166-y.
  * Aerts & Aerts (1995) Aerts, D. and Aerts, S. (1995). Applications of quantum statistics in psychological studies of decision processes. Foundations of Science 1, pp. 85-97. doi: 10.1007/BF00208726.
  * Aerts & Aerts Arguëlles (2022) Aerts, D. and Aerts Arguëlles, J. (2022). Human perception as a phenomenon of quantization. Entropy 24, 1207. doi: 10.3390/e24091207.
  * Aerts & Beltran (2020) Aerts, D. and Beltran, L. (2020). Quantum Structure in Cognition: Human Language as a Boson Gas of Entangled Words. Foundations of Science 25, pp. 755-802. doi: 10.1007/s10699-019-09633-4.
  * Aerts & Beltran (2022a) Aerts, D. and Beltran, L. (2022a). Are words the quanta of human language? Extending the domain of quantum cognition. Entropy 24, 6. doi: 10.3390/e24010006.
  * Aerts & Beltran (2022b) Aerts, D. and Beltran, L. (2022b). A Planck Radiation and Quantization Scheme for Human Cognition and Language. Frontiers in Psychology 13, 850725. doi: 10.3389/fpsyg.2022.850725.
  * Aerts et al. (2012) Aerts, D., Broekaert, J., Gabora, L. and Veloz, T. (2012). The guppy effect as interference. Quantum Interaction. Lecture Notes in Computer Science 7620, pp 36-47. doi: 10.1007/978-3-642-35659-9_4.
  * Aerts & Gabora (2005a) Aerts, D and Gabora, L. (2005a). A theory of concepts and their combinations I: The structure of the sets of contexts and properties.Kybernetes 34, pp. 167-191. doi: 10.1108/03684920510575799.
  * Aerts & Gabora (2005b) Aerts, D and Gabora, L. (2005b). A theory of concepts and their combinations II: A Hilbert space representation. Kybernetes 34, pp. 192-221. doi: 10.1108/03684920510575807.
  * Aerts & Sassoli de Bianchi (2014) Aerts, D. and Sassoli de Bianchi, M. (2014). The extended Bloch representation of quantum mechanics and the hidden-measurement solution to the measurement problem. Annals of Physics 351, pp. 975-1025. doi: 10.1016/j.aop.2014.09.020. See also the Erratum: Annals of Physics 366, pp. 197-198 (2016). doi: 10.1016/j.aop.2016.01.001.
  * Aerts & Sassoli de Bianchi (2015) Aerts, D. and Sassoli de Bianchi, M. (2015). The unreasonable success of quantum probability II: Quantum measurements as universal measurements. Journal of Mathematical Psychology 67, pp. 76-90. doi: 10.1016/j.jmp.2015.05.001.
  * Aerts & Sassoli de Bianchi (2016) Aerts, D. and Sassoli de Bianchi, M. (2016). The extended Bloch representation of quantum mechanics: Explaining superposition, interference, and entanglement. Journal of Mathematical Physics 57. 122110. doi: 10.1063/1.4973356.
  * Aerts & Sassoli de Bianchi (2017) Aerts, D. and Sassoli de Bianchi, M. (2017). Universal Measurements: How to Free Three Birds in One Move. Singapore: World Scientific.
  * Aerts et al. (2020) Aerts, D., Sassoli de Bianchi, M., Sozzo, S. and Veloz, T. (2020). On the conceptuality interpretation of quantum and relativity theories. Foundations of Science 25, pp. 5–54. doi: 10.1007/s10699-018-9557-z.
  * Aerts & Sozzo (2011) Aerts, D. and Sozzo, S. (2011). Quantum structure in cognition: Why and how concepts are entangled. Quantum Interaction QI2011. Lecture Notes in Computer Science 7052, pp. 116-127. doi: 10.1007/978-3-642-24971-6\\_12.
  * Aerts & Sozzo (2014) Aerts, D. and Sozzo, S. (2014). Quantum entanglement in concept combinations. International Journal of Theoretical Physics 53, pp. 3587–3603. doi: 10.1007/s10773-013-1946-z.
  * Aerts Arguëlles (2018) Aerts Arguëlles, J. (2018). The heart of an image: Quantum superposition and entanglement in visual perception. Foundations of Science 23, pp. 757-778. doi: 10.1007/s10699-018-9547-1.
  * Aerts Arguëlles & Sozzo (2020) Aerts Arguëlles, J. and Sozzo, S. (2020). How images combine meaning. Quantum entanglement in visual perception. Soft Computing 24, pp. 10277-10286. doi: 10.1007/s00500-020-04692-3.
  * Berlin & Kay (1969) Berlin, B. and Kay, P. (1969). Basic Color Terms: Their Universality and Evolution. Berkeley: University of California Press.
  * Blutner & beim Graben (2016) Blutner, R. and beim Graben, P. (2016). Quantum cognition and bounded rationality. Synthese 193, pp. 3239-3291. doi: 10.1007/s11229-015-0928-5.
  * Bruner & Postman (1949) Bruner, J. S. and Postman, L. (1949). On the perception of incongruity: A paradigm. Journal of Personality 18, pp. 206-223. doi: 10.1111/j.1467-6494.1949.tb01241.x.
  * Bruza & Gabora (2009) Bruza, P. and Gabora, L. (Eds.) (2009). Special Issue: Quantum Cognition. Journal of Mathematical Psychology 53, pp. 303-452. doi: 10.1016/j.jmp.2009.06.002.
  * Busemeyer & Bruza (2012) Busemeyer, J. and Bruza, P. (2012). Quantum Models of Cognition and Decision. Cambridge: Cambridge University Press.
  * Busemeyer et al. (2019) Busemeyer, J., Kvam, P. and Pleskac, T. (2019). Markov versus quantum dynamic models of belief change during evidence monitoring. Scientific Reports 9, 18025. doi: 10.1038/s41598-019-54383-9.
  * Busemeyer et al. (2006) Busemeyer, J. R., Wang, Z. and Townsend, J. T. (2006). Quantum dynamics of human decision making. Journal of Mathematical Psychology 50, pp. 220-241. doi: 10.1016/j.jmp.2006.01.003.
  * Collier at al. (1973) Collier, G. A., Berlin, B. and Kay, P. (1973). Basic color terms: Their universality and evolution. Language 49, pp. 245-248. doi: 10.2307/412128.ISSN0097-8507.
  * Dalla Chiara et al. (2015) Dalla Chiara, M. L., Giuntini, R., Leporini, R., Negri, E. and Sergioli, G. (2015). Quantum information, cognition, and music. Frontiers in Psychology 6, 1583. doi: 10.3389/fpsyg.2015.01583.
  * Davidoff (2001) Davidoff, J. (2001). Language and perceptual categorisation. Trends in Cognitive Sciences 5 pp. 382-387. doi: 10.1016/s1364-6613(00)01726-5.
  * Davies et al. (1998) Davies, I. R. L., Sowden, P. T., Jerrett, D. T., Jerrett, T. and Corbett, G. G. (1998). A cross-cultural study of English and Setswana speakers on a colour triads task: A test of the Sapir-Whorf hypothesis. British Journal of Psychology. 89, pp. 1-15. doi: 10.1111/j.2044-8295.1998.tb02669.x.
  * Dirac (1939) Dirac, P. A. M. (1939). A new notation for quantum mechanics. Mathematical Proceedings of the Cambridge Philosophical Society 35 pp. 416-418. doi: 10.1017/S0305004100021162.
  * Disa et al. (2011) Disa, S., LeGuen, O. and Haun, D. (2011). Categorical perception of emotional facial expressions does not require lexical categories. Emotion 11, pp. 1479-1483. doi: 10.1037/a0025336.
  * Eimas et al. (1971) Eimas, P. D., Siqueland, E. R., Jusczyk, P. W. and Vigorito, J. (1971). Speech perception in infants. Science 171, pp. 303-306. doi: 10.1126/science.171.3968.303.
  * Geeraerts et al. (2001) Geeraerts, D., Dirven, R., Taylor, J. R. and Langacker, R. W. (Eds.), (2001). Applied Cognitive Linguistics, II, Language Pedagogy. doi: 10.1515/9783110866254.
  * Goldstone & Hendrickson (2010) Goldstone, R.L. and Hendrickson, A.T. (2010). Categorical perception. Wires Cognitive Science 1, pp. 69–78. doi: 10.1002/wcs.26.
  * Gudder & Zanghi (1984) Gudder, S., Zanghí, N. Probability models. Nuovo Cimento B 79, pp. 291-301. doi: 10.1007/BF02748978.
  * Harnad (1987) Harnad, S. (Ed.). (1987). Categorical Perception: The Groundwork of Cognition. Cambridge, UK: Cambridge University Press.
  * Haven & Khrennikov (2013) Haven, E. and Khrennikov, A. (2013). Quantum Social Science. Cambridge: Cambridge University Press.
  * Johanden & Kruschke (2005) Johansen, M. K. and Kruschke, J. K. (2005). Category representation for classification and feature inference. Journal of Experimental Psychology: Learning, Memory, and Cognition 31, pp. 1433-1458. doi: 10.1037/0278-7393.31.6.1433.
  * Havy & Waxman (2016) Havy, M. and Waxman, S. R. (2016). Naming influences 9-month-olds’ identification of discrete categories along a perceptual continuum. Cognition 156, pp. 41-51. doi: 10.1016/j.cognition.2016.07.011.
  * Hess et al. (2009) Hess, U., Adams, R. and Kleck, R (2009). The categorical perception of emotions and traits. Social Cognition 27, pp. 320-326. doi: 10.1521/soco.2009.27.2.320.
  * Khrennikov (2014) Khrennikov, A. (2014). Ubiquitous Quantum Structure. Berlin: Springer.
  * Lane (1965) Lane, H. (1965). The motor theory of speech perception: A critical review. Psychological Review 72, pp. 275-309. doi: 10.1037/h0021986.
  * Lawrence (1949) Lawrence, D. H. (1949). Acquired distinctiveness of cues: I. Transfer between discriminations on the basis of familiarity with the stimulus. Journal of Experimental Psychology 39, pp. 770-784. doi: 10.1037/h0058097.
  * Liberman et al. (1967) Liberman, A. M., Cooper, F. S., Shankweiler, D. P., Studdert-Kennedy, M. (1967). Perception of the speech code. Psychological Review 74 pp. 431-461. doi:10.1037/h0020279.
  * Liberman et al. (1957) Liberman, A. M., Harris, K. S., Hoffman, H. S. and Griffith, B. C. (1957). The discrimination of speech sounds within and across phoneme boundaries. Journal of Experimental Psychology 54, pp. 358-368. doi:10.1037/h0044417.
  * Medin at al. (1984) Medin, D. L., Altom, M. W. and Murphy, T. D. (1984). Given versus induced category representations: Use of prototype and exemplar information in classification. Journal of Experimental Psychology: Learning, Memory, and Cognition 10, pp. 333-352. doi: 10.1037/0278-7393.10.3.333.
  * Mervis et al. (1975) Mervis, B. C., Catlin, J. and Rosch, E. (1975). Development of the structure of color categories. Developmental Psychology 2, pp. 54-60. doi: 10.1037/h0076118.
  * Moreira & Wichert (2016) Moreira, C. and Wichert, A. (2016). Quantum probabilistic models revisited: the case of disjunction effects in cognition. Frontiers in Physics 4. 26. doi: 10.3389/fphy.2016.00026.
  * Osherson & Smith (1981) Osherson, D. N. and Smith, E. E. (1981). On the adequacy of prototype theory as a theory of concepts. Cognition 9 pp. 35-58. doi:10.1016/0010-0277(81)90013-5.
  * Pothos et al. (2015) Pothos, E. M., Barque-Duran, A., Yearsley, J. M., Trueblood, J. S., Busemeyer, J. R. and Hampton, J. A. (2015). Progress and current challenges with the quantum similarity model. Frontiers in Psychology 6, 205. doi: 10.3389/fpsyg.2015.00205.
  * Regier & Kay (2009) Regier, T. and Kay, P. (2009). Language, thought, and color: Whorf was half right. Trends in Cognitive Sciences 13, pp. 439-447. doi: 10.1016/j.tics.2009.07.001.
  * Rosch (1973) Rosch, E. H. (1973). Natural categories. Cognitive Psychology 4, pp. 328-350. doi: 10.1016/0010-0285(73)90017-0.
  * Rosch Heider (1971) Rosch Heider, E. (1971). “Focal” Color areas and the development of color names. Developmental Psychology 4, pp. 447-455. doi: 10.1037/h0030955.
  * Rosch Heider (1972) Rosch Heider, E. (1972). Universals in color naming and memory. Journal of Experimental Psychology 93, pp. 10-20. doi: 10.1037/h0032606.
  * Rosch (1975) Rosch, Eleanor (1975). Cognitive representations of semantic categories. Journal of Experimental Psychology: General 104, pp. 192-233. doi: 10.1037//0096-3445.104.3.192. doi: 10.1037/h0032606.
  * Rosch et al. (1976) Rosch, E., Mervis, C. B, Gray, W. D, Johnson, D. M and Boyes-Braem, P. (1976). Basic objects in natural categories. Cognitive Psychology. 8, pp. 382-439. doi: 10.1016/0010-0285(76)90013-X.
  * Schusterman et al. (2000) Schusterman, R. J., Reichmuth, C. J. and Kastak, D. (2000). How animals classify friends and foes. Current Directions in Psychological Science 9, pp. 1-6. doi: 10.1111/1467-8721.00047.
  * Sidman (1994) Sidman M. (1994). Equivalence Relations and Behavior: A Research Story. Boston, MA: Authors Cooperative.
  * Smith & Medin (1981) Smith, E. E. and Medin, D. L. (1981). Categories and Concepts. Cambridge MA: Harvard University Press.
  * Surov et al. (2019) Surov, I. A., Pilkevich, S. V., Alodjants, A. P. and Khmelevsky, S. V. (2019). Quantum phase stability in human cognition. Frontiers in Psychology 10, 929. doi: 10.3389/fpsyg.2019.00929.
  * Yearsley (2017) Yearsley, JM. (2017). Advanced tools and concepts for quantum cognition: a tutorial. Journal of Mathematical Psychology 78, pp. 24-39. doi: 10.1016/j.jmp.2016.07.005.