# The Quantum Nature of Color Perception: Uncertainty Relations for Chromatic Opposition

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## Abstract

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## 1. Introduction

## 2. The Dawn of Hyperbolicity in Color Perception: Yilmaz’s and Resnikoff’s Works

#### 2.1. Yilmaz’s Relativity of Color Perception

#### 2.2. Resnikoff’s Homogeneous Color Space

- $\mathcal{C}$ is a cone if, for all $c\in \mathcal{C}$ and all $\lambda \in {\mathbb{R}}^{+}$, $\lambda c\in \mathcal{C}$, i.e., $\mathcal{C}$ is stable w.r.t. multiplications by a positive constant. This is the mathematical translation of the fact that, up to the glare limit, if we can perceive a color, then we can also perceive a brighter version of it;
- $\mathcal{C}$ is convex if, for every couple ${c}_{1},{c}_{2}\in \mathcal{C}$ and every $\alpha \in [0,1]$, $\alpha {c}_{1}+(1-\alpha ){c}_{2}\in \mathcal{C}$, i.e., the line segment connecting two perceived colors is entirely composed by perceived colors;
- $\mathcal{C}$ is regular if, denoted with $\overline{\mathcal{C}}$ its closure w.r.t. the topology induced by the inner product of V, the conditions $c\in \overline{\mathcal{C}}$ and $-c\in \overline{\mathcal{C}}$ imply that $c=0$. The intuitive geometrical meaning of this condition is that $\mathcal{C}$ is a single cone with a vertex.

## 3. Jordan Algebras and Their Use in Quantum Theories

#### 3.1. Basic Results and Classification of Three-Dimensional Formally Real Jordan Algebras and Their Positive Cones

#### 3.2. Jordan Algebras and Algebraic Formulation of Quantum Theories

- A physical system $\mathcal{S}$ is described as a setting where we can perform physical measurements giving rise to quantitative results in conditions that are as isolated as possible from external influences;
- A state of $\mathcal{S}$ is the way it is prepared for the measurement of its observables;
- Observables in $\mathcal{S}$ are the objects of measures and are associated with the physical apparatus used to measure them on a given state;
- An expected value of an observable in a given state is the average result of multiple measurements of the observable when the system is prepared each time in the same state.

- A physical system $\mathcal{S}$ is described by its observables, which are elements of an algebra $\mathcal{A}$ with unit $\mathbf{1}$ endowed with a partial ordering. Notice that this does not mean that all the elements of $\mathcal{A}$ are observables, but only that the observables of $\mathcal{S}$ are contained in $\mathcal{A}$;
- If $\mathcal{A}$ is a commutative and associative algebra, then we deal with a classical system; otherwise, we call $\mathcal{S}$ a quantum system;
- A state on $\mathcal{A}$ is a positive normalized linear functional $\omega :\mathcal{A}\to \mathbb{R}$, i.e.,
- -
- If $a\in \mathcal{A}$ is positive, accordingly to the partial ordering of $\mathcal{A}$, then $\omega \left(a\right)\ge 0$;
- -
- $\omega \left(\mathbf{1}\right)=1$.

- Given an observable $a\in \mathcal{A}$ and a preparation of the system, i.e., a state $\omega $, we can associate the number ${\langle a\rangle}_{\omega}:=\omega \left(a\right)$, called the expectation value of the variable $a\in \mathcal{A}$ on the state $\omega $. $\omega \left(a\right)$ is operationally obtained by performing replicated measurements of a on identically prepared states and by taking the average over the outcomes of measurements.

- $\rho $ describes a pure state if and only if Tr$\left({\rho}^{2}\right)=1$;
- $\rho $ describes a mixed state if and only if Tr$\left({\rho}^{2}\right)<1$.

- If $\mathcal{A}$ is a real matrix Jordan algebra, then$$S\left(\rho \right)=-\sum _{k}{\lambda}_{k}log{\lambda}_{k},$$
- $\rho $ describes a pure state if and only if $S\left(\rho \right)=0$ (minimum entropy, i.e., maximal amount of information), if $S\left(\rho \right)>0$, $\rho $ describes to a mixed state;
- S is invariant under orthogonal conjugation, i.e., $S\left(O\rho {O}^{t}\right)=S\left(\rho \right)$, for all $O\in \mathrm{O}\left(n\right)$, $n=dim\left(\mathcal{A}\right)$;
- $S\left(\rho \right)$ is a concave function of $\rho $ in the following sense: $S(t{\rho}_{1}+(1-t){\rho}_{2})\ge tS\left({\rho}_{1}\right)+(1-t)S\left({\rho}_{2}\right)$ for all couples of density matrices ${\rho}_{1}$, ${\rho}_{2}$.
- The state of maximal entropy, or of minimal amount of information, is the normalized unit of the FRJA $\mathcal{A}$:$${\rho}_{0}:=\underset{\rho}{\mathrm{argmax}}\phantom{\rule{0.277778em}{0ex}}S\left(\rho \right)=\frac{1}{\mathrm{Tr}(1)}\mathbf{1}\in \mathcal{DM}\left(\mathcal{A}\right).$$

## 4. A Quantum Theory of Color Perception

- A visual scene is a setting where we can perform psycho-visual measurements in conditions that are as isolated as possible from external influences;
- A perceptual chromatic state is represented by the preparation of a visual scene for psycho-visual experiments;
- A perceptual color is the observable identified with a psycho-visual measurement performed on a given perceptual chromatic state;
- A perceived color is the expected value of a perceptual color after a psycho-visual measurement.

- Well-known colorimetric definitions such as additive or substractive synthesis of color stimuli, aperture or surface color, color in context, and so on, are incorporated in the concept of preparation of a perceptual chromatic state. A first example of preparation is the set up a visual scene where an observer in a dark room has to look at a screen, where a light stimulus with foveal aperture w.r.t. the observer provokes a color sensation. A second example of preparation is given by an observer adapted to an illuminant in a room who looks at the patch of a surface. The perceptual chromatic states identified by these two preparations are, in general, different;
- The instruments used to measure the observables are not physical devices, but the sensory system of a human being. Moreover, the results may vary from person to person, thus the average procedure needed to experimentally define the expected value of an observable on a given state is, in general, observer-dependent. The response of an ideal standard observer can be obtained through a further statistical average on the observer-dependent expected values of an observable in a given state.

- A theory of color perception associated to the FRJA $\mathbb{R}\oplus \mathbb{R}\oplus \mathbb{R}$ is classical;
- A theory of color perception associated to the FRJAs $\mathcal{H}(2,\mathbb{R})\cong \mathbb{R}\oplus {\mathbb{R}}^{2}$ is quantum-like.

#### 4.1. Pure and Mixed Quantum Chromatic States

#### 4.2. Von Neumann Entropy of Quantum Chromatic States: Saturation and Hue

#### 4.3. Hering’s Chromatic Opponency and Its Role in the Encoding of Visual Signals

- The maximally von Neumann entropy state ${\rho}_{0}$, which represents the achromatic state;
- Two pairs of diametrically opposed pure hues.

#### 4.4. Uncertainty Relations for Chromatic Opponency

**Proposition**

**1.**

**Proof.**

- The only value of r that nullifies the right hand side of (60) is $r=0$, that corresponds to an achromatic color state, thus no uncertainty about the degree of opposition R-G and Y-B is present in this case. This fact is coherent with both our physiological and perceptual knowledge of color perception: a perceived achromatic color is characterized by an ‘equal amount of chromatic opponencies’;
- The only values of $\vartheta $ that nullify the right hand side of (60) are $\vartheta =0,\frac{\pi}{2},\pi ,\frac{3}{2}\pi $, which identify the two opposition axes R-G and Y-B. Again, this seems coherent with common knowledge: suppose that we want to determine the couple of opponencies R-G and Y-B to match, say, a color perceived as a red but with a non maximal saturation (measured as a function of the von Neumann entropy of its chromatic state). Due to the redness of the percept, we will always set an equal opposition in the axis Y-B that will not influence the search for the correct opposition R-G. Thus, the determination of the two chromatic oppositions will be compatible;
- Instead, for $r\in (0,1]$ and $\vartheta \in [0,2\pi )\setminus \{0,\frac{\pi}{2},\pi ,\frac{3}{2}\pi \}$, there will be a lower bound strictly greater than 0 for the product of quadratic dispersions of ${\sigma}_{1}$ and ${\sigma}_{2}$ on the state defined by $\rho (r,\vartheta )$. Moreover, this lower bound is a non-linear function of the variables $(r,\vartheta )$ and it is maximum for pure hues, $r=1$, halfway in between G and B, B and R, R and Y and Y and G, i.e., $\vartheta =\frac{\pi}{4}+k\frac{\pi}{2}$, $k=1,2,3$. If this interpretation is correct, then trying to adjust the R-G opposition to match, say, a color perceived as orange, should introduce a ‘perceptual disturbance’ on the adjustment of the opposition Y-B.

#### 4.5. Geometry and Metrics of Quantum Chromatic States

## 5. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**An example background change: in spite of the fact that the inner disks in the center of the two images provide exactly the same physical color stimulus, our perception of them is different because of the so-called chromatic induction phenomenon due to the background difference.

**Figure 2.**The crispening effect showing that Resnikoff’s assumption about the invariance of color metric with respect to background transformations is not coherent with perception.

**Figure 3.**A proposal for the saturation of a quantum chromatic state from built from its von Neumann entropy.

**Figure 4.**The unit (Bloch) disk $\mathbf{D}$ in ${\mathbb{R}}^{2}$, whose points represent quantum chromatic states. The points of its border ${S}^{1}$ represent pure quantum chromatic states and the center represents the achromatic state.

**Figure 5.**The polar graph of the lower bound $\frac{{r}^{4}}{4}{sin}^{2}\left(2\vartheta \right)$ of Equation (60). Notice that this lower bound is 0 only for points with coordinates $(r,\vartheta )$ lying on the opponency axes, identified with the horizontal and vertical axes in the picture. The lower bound is maximal for $(r,\vartheta )=(1,\frac{\pi}{4}+k\frac{\pi}{2})$, $k=1,2,3$.

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Berthier, M.; Provenzi, E.
The Quantum Nature of Color Perception: Uncertainty Relations for Chromatic Opposition. *J. Imaging* **2021**, *7*, 40.
https://doi.org/10.3390/jimaging7020040

**AMA Style**

Berthier M, Provenzi E.
The Quantum Nature of Color Perception: Uncertainty Relations for Chromatic Opposition. *Journal of Imaging*. 2021; 7(2):40.
https://doi.org/10.3390/jimaging7020040

**Chicago/Turabian Style**

Berthier, Michel, and Edoardo Provenzi.
2021. "The Quantum Nature of Color Perception: Uncertainty Relations for Chromatic Opposition" *Journal of Imaging* 7, no. 2: 40.
https://doi.org/10.3390/jimaging7020040