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The Journal of Mathematical Neuroscience

Geometry of color perception. Part 2: perceived colors from real quantum states and Hering’s rebit

M. Berthier

Published Sep 9, 2020• 25 citations
Publisher

Abstract

Inspired by the pioneer work of H.L. Resnikoff, which is described in full detail in the first part of this two-part paper, we give a quantum description of the space [Formula: see text] of perceived colors. We show that [Formula: see text] is the effect space of a rebit, a real quantum qubit, whose state space is isometric to Klein's hyperbolic disk. This chromatic state space of perceived colors can be represented as a Bloch disk of real dimension 2 that coincides with Hering's disk given by the color opponency mechanism. Attributes of perceived colors, hue and saturation, are defined in terms of Von Neumann entropy.

DOI
10.1186/s13408-020-00092-x

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Inspired by the pioneer work of H.L. Resnikoff, which is described in full detail in the first part of this two-part paper, we give a quantum description of the space [Formula: see text] of perceived colors. We show that [Formula: see text] is the effect space of a rebit, a real quantum qubit, whose state space is isometric to Klein's hyperbolic disk. This chromatic state space of perceived colors can be represented as a Bloch disk of real dimension 2 that coincides with Hering's disk given by the color opponency mechanism. Attributes of perceived colors, hue and saturation, are defined in terms of Von Neumann entropy.
In this paper, we provide an overview on the foundation and first results of a very recent quantum theory of color perception, together with novel results about uncertainty relations for chromatic opposition. The major inspiration for this model is the 1974 remarkable work by H.L. Resnikoff, who had the idea to give up the analysis of the space of perceived colors through metameric classes of spectra in favor of the study of its algebraic properties. This strategy permitted to reveal the importance of hyperbolic geometry in colorimetry. Starting from these premises, we show how Resnikoff's construction can be extended to a geometrically rich quantum framework, where the concepts of achromatic color, hue and saturation can be rigorously defined. Moreover, the analysis of pure and mixed quantum chromatic states leads to a deep understanding of chromatic opposition and its role in the encoding of visual signals. We complete our paper by proving the existence of uncertainty relations for the degree of chromatic opposition, thus providing a theoretical confirmation of the quantum nature of color perception.
Starting from the foundational axiomatization of the perceptual color space initiated by Schrödinger in 1920 and eventually refined by Resnikoff in 1974, Berthier, Provenzi and their collaborators have recently proposed a reformulation of perceptual color attributes within the framework of quantum information. Their work is based on the Jordan algebra formalism of quantum theories and, more specifically, on a quantum system described by a spin factor over the field of real numbers. This theoretical framework is not that of ordinary quantum mechanics, mainly because it requires dealing with rebits, whereas the latter uses qubits. The aim of this paper is to show that this difference in no way hinders the implementation of experimental protocols for testing the validity of the predictions of the color perception model. In particular, we show how to compute the quantum information based perceptual attributes of perceived colors in terms of qubit density matrices.
In this paper we make a systematic use of the quantum measurement theory to describe perceived colors and analyze some of their fundamental properties. After motivating the naturalness of the quantum measurement approach in the mathematical framework of the color perception theory proposed by the authors in previous papers, we show how to obtain several results. Among our theoretical outcomes, we mention the possibility to confine the color cone to a finite-volume color solid and the link between post-measurement state changes, Lorentz boosts and the Einstein-Poincaré relativistic addition law. We apply these results to obtain a chromatic match equation that emphasizes the importance of the Hilbert-Klein metric on the unit disk and we also present a quantitative description of Hunt's effect.
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In this paper we deal with the problem of overcoming the intuitive definition of several color perception attributes by replacing them with novel mathematically rigorous ones. Our framework is a quantum-like color perception theory recently developed, which constitutes a radical change of view with respect to the classical CIE models and their color appearance counterparts. We show how quantum information concepts, as e.g. effects, generalized states, post-measurement transformations and relative entropy provide tools that seem to be perfectly fit to model color perception attributes as brightness, lightness, colorfulness, chroma, saturation and hue. An illustration of the efficiency of these novel definitions is provided by the rigorous derivation of the so-called lightness constancy phenomenon.
In this paper, we make a systematic use of the quantum measurement theory to describe perceived colours and analyse some of their fundamental properties. After motivating the naturalness of the quantum measurement approach in the mathematical framework of the colour perception theory proposed by the authors in previous papers, we show how to obtain several results. Among our theoretical outcomes, we mention the possibility to confine the colour cone to a finite-volume colour solid and the link between post-measurement state changes, Lorentz boosts and the Einstein–Poincaré relativistic addition law. We apply these results to obtain a chromatic match equation that emphasizes the importance of the Hilbert–Klein metric on the unit disk and we also present a quantitative description of Hunt’s effect.