Probabilities, with which an individual chooses basis alternatives from the experiential state (Equation 1), are defined by standard quantum-theoretic Born's rule

and measured statistically for an ensemble of identically (indistinguishably) staged experiments. In the above expression, 〈ψ|i〉 denotes the inner product of basis vectors |0〉, |1〉 with complex-conjugate (Hermitian) transpose of state vector (Equation 2)

Decision probabilities 0 ≤ p_i ≤ 1 quantify conduciveness of the represented context for the basis decision alternatives as considered by a subject. Angle θ thus functions as the evaluative dimension of the qubit state (Equation 1), shown in Figure 2A. The top and bottom Bloch hemispheres in Figure 1 then accommodate representations of positively and negatively evaluated contexts, respectively.

Consider a decision whether to have a cup of tea (1) or not (0). It generates a task-specific Hilbert space of individual experience of a decision maker, accommodating subjective representations of all possible decision contexts. In this space, the context described, e.g., by a single word hurry is likely to be mapped to the qubit state (Equation 1)

Since hurry is not conducive for tea parties, polar angle θ must be less than 90°, so that pno tea=cos2(θ/2)>sin2(θ/2)=ptea.

As a typical result of having tea, the context relaxation, analogously, is likely to be mapped to the upper Bloch hemisphere with θ > 90°. The context dog, in contrast, has no conventional relation with tea. If an individual had no personal reason to establish one, the representation will fall near the equator of the sphere (or near its equatorial section, as further discussed in Section 2.4).

In line with the overall discreteness of human cognition (Zipf, 1945; Rosch, 1973; Tee and Taylor, 2020), the process dimension is divided into distinct stages, facilitating recognition and categorization tasks. The number of stages is a matter of convenience, varying for different applications. Following (Surov, 2022), this article uses three primary and three secondary stages including perception, novelty, goal-plan, action, progress, and result. The resulting six-part structure, dividing the azimuthal angle ϕ into equal ranges of 60° each, is shown in Figure 2B.

After discretization, the qubit's azimuthal dimension functions as a process-semantic template for subjective representation of contexts. Since favorability is defined only for specific content, this structure itself is evaluatively-neutral. Allocation of the contexts in this template then constitutes a causal model, imposed by a subject on them relative to the basis decision alternative (Cheng, 1997; Hubbard, 2012).

Continuing the tea-making case, the contexts are now allocated to different process stages in addition to their favorability considered above. In Equation (5) these process-semantic classes specify the phase angle ϕ according to the scheme shown in Figure 2B. Namely,

Besides this choice and mapping of contexts, of course, many others are possible. The same tea, for example, might be used to facilitate a conversation. Thirst and flavor then may be of minor significance and the same process stages would be populated by different contexts. In each case, the whole cycle constitutes a causal model of selected contexts, constructed by the subject of the basis decision alternative.

As seen from the above examples, cognitive algorithms used to construct qubit states can be quite complex. For an other basis, e.g., to take a medicine (1) or not (0), mapping of similar contexts would require very different knowledge. The diversity of human behavior and complexity of the corresponding cognition thereby makes context mapping for each decision basis largely unique.

In elementary quantum-physical systems, in contrast, the context-mapping algorithms are the same for all practiced decisions, often allowing compact analytical expression (Feynman et al., 1964, ch. 5, 6). Accordingly, knowing a qubit state of a “spin-1/2” particle in any given basis allows physicists to find its state for any other basis by corresponding rotation of the Bloch sphere. Whether some analogous procedure is possible for macroscopic individuals is an open question.

According to Section 2.1, the model considers emotional experiences as defined by the evaluative dimension θ and the process-semantic dimension ϕ. Major emotions (Tomkins and Mccarter, 1964; Izard, 1977; Lazarus, 1991; Ekman and Davidson, 1994)

then label classes of qubit states, defined by ranges of these two dimensions.

In particular, joy and sadness are most appropriate at the Result stage and not at Novelty or Action, when there is nothing yet to estimate; acceptance and disgust are experienced at the Progress stage as a feedback to the previous Action. The Action itself, in contrast, is facilitated by highly energetic aspiration, passion, zeal, and rage-type emotions, inappropriate for estimation of the Result or recognition of a Novelty. Instead, the latter is accompanied by curiosity, interest, confusion, startle, or fear. Each process stage thereby defines a class of emotional experiences, including both positive and negative ones.

In each process-semantic class, positive and negative emotions are discriminated by the polar angle θ, occupying the top (θ > 90°) and bottom (θ < 90°) halves of the Bloch sphere, respectively. For example, anxiety, startle, fear, and horror are increasingly negative experiences of Novelty, differentiated by polar angle decreasing from near the equator θ = 90° toward the south pole θ = 0°. For presentation, all such negative experiences of Novelty-class contexts with ϕ ≈ 60° are assembled into a single emotional prototype of fear.

The resulting map of prototype emotions to the Bloch sphere is shown in Figure 3. The primary process Novelty, Action, and Result generate six major emotional classes fear-surprise, anger-zeal, and sadness-joy, previously identified by Shaver et al. (1987). Secondary stages, shown in gray, are less expressive so that the corresponding emotional terms are lower in typicality rating (Fehr and Russell, 1984).

The spherical geometry of the qubit model generalizes well-known circular models of emotion. The most popular valence-arousal circumplex (Russell, 1980; Barrett and Russell, 1998), in particular, is a section of the Bloch sphere by the Y-Z plane. These and similar models (Hevner, 1936; Schlosberg, 1952; Scherer et al., 2013), thus, appear as empirically discovered marginals of the qubit model of subjective experience (Surov, 2022).

Parameter c in Equation (7) is coherence of the qubit state (Baumgratz et al., 2014). It measures an ability of an individual to affect the resolution of the basis decision alternative, which may be called subjectness or subjective freedom. Full freedom corresponds to pure states (Equation 1) with maximal value c = 1/2, achieved for neutrally-evaluated experience with p₀ = p₁ = 1/2. In this case, contexts are mapped to the equator of the Bloch sphere, allowing for maximal resolution of the process-semantic stages.

Zero freedom c = 0 means that this alternative is either already resolved, or that the considered subject is unable to (meaningfully) affect its resolution (for example, due to the lack of appropriate process-semantic model, structuring the contexts). Accordingly, subjective representation of the contexts by a pure state is either useless or impossible due to limitations of an individual. In both cases, representation space reduces to the diameter of the Bloch sphere, visualizing classical binary uncertainty (Aerts, 1999; Surov, 2021).

Intermediate regimes of limited coherence are modeled by squeezing the Bloch sphere by factor c in X and Y directions. The resulting cognitive-semantic spaces are visualized by ellipsoids of rotation, built on the same poles |0〉 and |1〉.

The uncertainty is resolved by a decision act, in which one of the previously superposed potential states comes to being, while the other is irreversibly discarded. Maintenance of the qubit representation is of no more use; the experiential space collapses with all involved contexts ceasing to make sense for the considered individual and basis. This process, known in physics as collapse of the wavefunction (Jaeger, 2017), is modeled by sending the coherence of the qubit state (Equation 7) to zero, as shown in Figure 2A. The result is a fully mixed state

describing ignorance of an already existing experimental outcome equivalently to the classical two-event probability space (Kolmogorov, 1956).

For the example of Sections 2.1 and 2.2, mixture (Equation 9) describes probabilistic guess (i) of a person ignorant of whether the tea party happened or not in the past, and also (ii) of a person with no control over the resolution of the same uncertainty (Equation 5) in the future.

By occupying the interior of the Bloch ball along with its surface, mixed states of subjective experience are encoded by three parameters, adding coherence c to spherical angles θ and ϕ. Equivalently, mixed states are defined by components x, y, and z of Stokes vector (Equation 8). These coordinates correspond to orthogonal Cartesian axes, shown in Figure 1, which also have definite cognitive functions.

Z component of Stokes vector (Equation 8) is uniquely defined by decision probabilities

Geometrically, point z is the projection of pure (Equation 1) and mixed (Equation 8) state vectors to the diameter of the Bloch sphere, with the resulting segment's lengths defining probabilities p₀ and p₁ according to Figure 2A. In particular, probability p₁ = (z + 1)/2 is minimal for contexts with z < 0, favoring outcome |0〉, and maximal for contexts with z > 0, favoring |1〉. Accordingly, the Z axis generalizes the evaluative function of polar angle θ to both pure and mixed experiences.

Cognitive function of the X and Y axes is defined by their position in the process cycle, shown in Figure 2B. Y-axis quantifies subjective activity of contexts, which is maximal at Action and minimal at Perception process stage. X-axis quantifies subjective freedom, openness, or potency of the context, defining how strongly its variation would influence subsequent process stages. It is maximal for contexts describing subjective Goals and minimal just before seeing the Result when the process is maximally predetermined.

Z, X, and Y axes are thus identical to evaluation, potency, and activity factors of classical semantics (Osgood, 1962; Osgood et al., 1975; Surov, 2022).

Cross-cultural associative stability of colors ascends to the basic environmental factors and the practical significance they represent. Red (R), yellow (Y), green (G), and blue (B), in particular, are abstracted from natural prototypes of fire-blood, the Sun, vegetation, and sky-water, respectively (Wundt, 1897; Wierzbicka, 1990). These colors, however, are secondary with respect to (macro) white and (macro) black, forming a primary distinction pair (Wierzbicka, 1990). Due to the stability and psychologically-experiential nature of most associations, color is identified as an integral part of an archetypal system of affective meaning, underlying cognition and behavior of humans (Wierzbicka, 1990; Yanshin, 1996, 2017; Borisova, 1997; Serov, 2004; Bazyma, 2005).

More intuitively, three-dimensional color space is formalized in the hue—saturation—lightness (HSL) model (Guilford, 1940; Levkowitz and Herman, 1993; Tian-Yuan, 1995; Ibraheem et al., 2012). Hue and saturation are polar coordinates in the tension—temperature plane as shown in Figure 4. Saturation is the radial distance from the achromatic lightness (Z) axis, quantifying the proximity of color experience to a gray of the same lightness. Hue is an angular dimension, the circularity of which is also established experimentally (Odbert et al., 1942; Helm, 1964; Bonnardel and Pitchford, 2006), confirming the intuition behind chromatic color circles. Hue, saturation, and lightness thus are cylindrical coordinates of the Cartesian system just mentioned, with white and black bases of the cylinder collapsed to single polar points as shown in Figure 4.

In its basic idea, the HSL model does not restrict the number and location of the main chromatic colors. Whether three, four, or Newton's seven, all of them find a place at the equator of the color solid, discretizing it accordingly. The bi-cone topology of the HSL model shown in Figure 4 is, therefore, commonly considered a proper representation of color experience in human cognition (Churchland, 2007).

With six basic colors, the HSL model is recently used to develop its quantum-inspired version, QHSL (Yan et al., 2021c). By using the qubit as a representational structure, the model connects color to the methods of quantum information, applying them to the image processing tasks (Yan et al., 2021c). QHSL, however, focuses on the formal side of the color representation without considering its semantic function, central to the present study. In this respect, the models are further compared in Section 4.4.2.

This correspondence, however, has a drawback that complicates building a quantitative model of color semantics on this basis. By placing yellow on the same horizon as red, green, and blue, this model ascribes it to the same level of evaluation and lightness. The falsity of this is clearly experienced after several minutes of work with colored text. In fact, yellow (RGB=110) makes a visibly weaker contrast with white (111), as compared to red (100), green (010), and blue (001). Quantitatively, the perceptual lightness of yellow amounts to 8 out of 9, whereas red, green, and blue range from 4 to 5 (Berlin and Kay, 1975, p.8); similar results are reported in Boynton and Olson (1987), Hardin (1988), Izmailov and Sokolov (1991).

The reason for this difference is revealed by inspection of the natural prototypes and associations of colors summarized in Section 3.1. In contrast to red, green, and blue, associations of yellow (Table 1) are almost entirely positive as befits its natural prototype, the Sun. The same positivity is typical for white, to which yellow is also close semantically: things called white like bread, milk, and skin are yellowish in fact. The color temperature of the Sun (near 6,000°), accordingly, is that of white light, seemingly dropping to yellow only near sunrises and sunsets.

The same difference in lightness is readily observed for cyan (RGB=011) and magenta (101). In the standard HSL model of color shown in Figure 4, therefore, hue is orthogonal to lightness formally but not experientially. While acceptable for color-coding, introduction of color meanings as made above, and illustrative use, a reliable representation of color semantics needs a more accurate basis developed below.

The qubit model provides insight into the possible decomposition of colors. Unlike standard approaches noted in Section 3.2, the number of basic colors appears to be fixed by the mathematics in use.

Table 2 is visualized by overlapping the qubit-color map, Figure 6, with the qubit-emotion sphere, Figure 3. The resulting map is shown in Figure 7. This is a planar layout of the Bloch sphere, with the azimuthal dimension ϕ categorized into six process-semantic sectors as before. In this view, each sector becomes a leaf-shaped area of width π/3*sinθ, defined by spherical geometry.

In terms of color, this is an area-preserving version of Figure 6B with reversed radial dimension. Accordingly, the south pole |0〉—black color and negative limit of all emotional classes—locates in the center, while the north pole |1〉—white color and positive limit of emotions—maps to the periphery. Process-stage labels Perception, Novelty, Goal-plan, Action, Progress, and Result are placed in their respective leafs near the equatorial circle θ = 90° due to their evaluative neutrality.

Parallel demarcation of the qubit state in categories of emotion and color, both being genuinely subjective, semantic-class phenomena (Alcaro et al., 2017; Yanshin, 2018), suggests asking which one is more fundamental. In contrast to emotion, typically associated with psychology of complex organisms, color has an objectively-measured physical primitive—a wavelength of light. This nominates color as the primary semantic code of Nature (Yanshin, 1996), subsequently hardwired into the physiology of vision and neural architectures of various forms of life (Sokolov, 2001a,b). Monochromatic, dichromatic, tetrachromatic, and more complex vision systems then may be seen as Nature's attempts to develop and use alternative structures of meaning.

The obtained correspondence between color and emotional classes has a superficial similarity with popular wheel-type schemes, ascribing basic emotions to specific positions in the color circle (Karimova, 2021). Plutchik's emotion solid, in particular, is close to the present model in mapping emotional states to a three-dimensional half-sphere structure with azimuthal sectors colored in natural order (Plutchik, 1958, 2001). Although superficially similar, this mapping is conceptually different from the present article.

The main contrast is the location of basic emotions in the azimuthal dimension. With shaky logic behind it, Plutchik's ordering (disgust—expectancy—anger—joy— acceptance—surprise—fear—sadness) has little agreement with the experiment (Plutchik, 1980). The present approach explains this by disparate nature of four basic opposites, underlying this order (Kellerman, 2020; ch. 5). Namely, joy-sadness and acceptance-disgust pairs are opposite in valence within the same process-stage classes, while anger-fear and expectancy-surprise differ in process stage rather than in valence. Accordingly, qubit representation distinguishes these states in different spherical coordinates instead of a single circular dimension.

In agreement with the semantics of colors (Section 3.1) this allows, in particular, to map fear to (dark) blue and sadness to (dark) green, in contrast to the reverse order postulated in Plutchik (2001). The scheme shown in Figure 7 is thus more coherent factually, conceptually, and mathematically. Plutchik's model, however, takes credit as an inspiring conjecture used in several research directions (Wang and Pereira, 2016; Gu et al., 2019; Yan et al., 2021b).

The developed model relates to the space of color-emotion categories proposed by Kozlovskiy et al. (2016) and Kiselnikov et al. (2019), also establishing correspondence between the two domains. Unlike the qubit model, however, this approach maps hue, saturation, and lightness to three angular dimensions, locating emotions and colors on the surface of a four-dimensional sphere (Izmailov and Sokolov, 1992; Sokolov and Boucsein, 2000; Leonov and Sokolov, 2008).

Although different geometrically and semantically, this representation is close to the qubit model in a crucial aspect of conceptual significance. Sokolov's color-emotion sphere is considered as a special instance of a fundamental cognitive architecture, encoding neuronal states by vectors of constant length. The neural basis of trichromatic vision just exemplifies this universal coding, central to other functions of human intelligence (Izmailov et al., 1989; Sokolov, 2000, 2001a). The present approach is of similar generality, at the same time introducing conceptually new aspects, unique to quantum semantics (Surov, 2022).

Psychological studies of emotion and color lack conceptually-quantitative expression, connecting them to observable phenomena. The resulting encapsulation in the cognitive realm deprives the models of practical application, undermining their fundamental value. The obtained result addresses this problem by situating emotion and color in the broader scope of behavioral semantics.

Color, in particular, appears as the natural encoding of qubit-semantic states, affecting objectively measurable regularities of decision making, quantified by the quantum probability theory. This setting resonates with the goals pursued in robotics and artificial intelligence (Breazeal, 2003; Cavallo et al., 2018; Pessoa, 2019b; Samsonovich, 2020b; Deng et al., 2021; Kotov et al., 2021; Yan et al., 2021a,b; Samsonovich, 2020a; ch. 7, 10, 13, 34, 54, 56, 57): as in natural cognitive systems, subjective states of individuals, essentially, couple stimuli to responses in a personally meaningful way, as required for non-trivial behavior. The established equivalence of color and emotion encodings of subjective meaning strengthens this practical approach. Emotion coloring (Izard, 1977; Pinker, 2008; Russell, 2009; Khrennikov, 2021), social color, color of information excitations (Khrennikov, 2018; Khrennikov et al., 2018), and money (Ferraris, 2019; Orrell, 2021), in particular, turn from insightful metaphors to rigorous mathematical correspondence, throwing light on empirically-discovered phenomena of practical interest (Bellizzi and Hite, 1992; Kwallek et al., 1996; Aslam, 2005; Hill and Barton, 2005; Doyle and Bottomley, 2006; Elliot et al., 2007; Labrecque et al., 2013; Neville, 2022).

Productive steps in this direction are, in fact, already made in quantum models of cognition and decision, developed in recent decades. In many cases, the correspondence is established simply by changing terminology from “cognitive” to “affective” states. Applied emotion science is then recognized in the existing models of cognitive fallacies and irrational decision-making (Busemeyer et al., 2011; Pothos and Busemeyer, 2013; Ashtiani and Azgomi, 2015), subjective utility (Basieva et al., 2018), non-classical social and economic behavior (Khrennikov, 2016; Njegovanovic, 2018; Meghdadi et al., 2022), semantics of natural language and information retrieval (Aerts et al., 2013; Melucci, 2015; Surov et al., 2021), conceptual and belief networks (Gabora et al., 2008; Moreira et al., 2020), cybernetics, artificial intelligence, and knowledge representation (Wolff et al., 2018; Bickley et al., 2021; Deng et al., 2021; Yan et al., 2021a). Color-emotion research, thus, gets a natural connection to vast areas of applied science and technology, at the same time acquiring the requested conceptual and theoretical basis (Brower, 1949; de Gelder, 2017; Reisenzein, 2019; Burghardt and Bodansky, 2021; Mascolo, 2021; Uher, 2021).

As inherited from physics (Svozil, 2018), a distinctive feature of quantum cognition is the lack of commonly accepted interpretation. The approach largely develops in a mathematically-formal way, detached from physical and psychological perspectives—a sort of theoretical black box, producing observable probabilities for particular behavioral cases without explaining them (Blutner and beim Graben, 2016; Sozzo, 2019). Although methodologically safe, this stance hinders the progress of the field, where productive models are often found by blind search and typically lack the predictive power, needed for practical use.

The obtained result suggests an approach to this problem. Color interface, for example, allows using standard methods of psychological diagnostics to find qubit's phase parameters in quantum models of cognition and decision, necessary to use them in predictive mode (Surov et al., 2019; Surov, 2021; Shan, 2022). Besides application to qubit-state models, the established color-emotion structure can facilitate interpretation of more complex wavefunctions, observables, projection and transformation operators, and other mathematical instruments, often having no clear psychological counterpart. This would open access to other results and methods of cognitive science and psychology, continuing a long history of productive interaction (Khrennikov, 2015).