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Markovian dynamics of single-rebit open quantum systems and information-theoretic analysis of color perception

Berthier, Michel, Gabriel Niebel, Edoardo Provenzi

Published Apr 22, 2025• 0 citations
OpenAlex

Abstract

This paper is devoted to the study of the Markovian dynamics of open two-state quantum systems defined over the real numbers. Its aim is twofold. On the one hand, we provide a complete mathematical description of such dynamics when the affine component of the quantum channel semigroup is Markovian. We describe Lindblad and Kraus operators, orthogonal evolutions by means of Jordan algebra derivations, as well as solutions of the GKSL dissipative equation and their effect on the relative entropy. On the other hand, we are interested in an original application to color perception that motivates this analysis. Using a recent novel model in which perceived colors are given by Lüders measurements on the state space of a rebit, we show how the chromatic distortion induced by a non-neutral illuminant can be modeled by a Markovian rebit channel that progressively reduces color distinguishability. This colorimetric effect is illustrated through simulations on digital images, offering a concrete demonstration of the relevance of Markovian rebit evolution for modeling perceptual phenomena.

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Markovian dynamics of single-rebit open quantum systems with applications to colour perception

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These notes are a short introduction to the mathematical theory of open quantum systems. They are meant to serve as an entry point into a broad research area which has applications across the quantum sciences dealing with systems subjected to external noise. The guiding idea is to let the key structures of the theory emerge from a concrete model. By working through the dissipative Jaynes-Cummings model the reader will dis- cover explicitly how irreversible dynamics arises from a unitary system-reservoir evolution. The notions of the continuous mode limit, correlation functions, spectral density appear in a natural manner and lead to the evolution equation of the open system in form of a master equation. This sets the stage for the more general analysis of completely positive, trace preserving (CPTP) maps and the study of quantum dynamical semigroups. We motivate and prove the Kraus representation theorem, the dilation theorem and the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) theorem. Working through the exercises (for which full solutions are supplied) will reinforce the ideas introduced in the main text.

quant-pharXiv:2603.11925•Mar 13, 2026•0 citations
This paper is devoted to the study of the Markovian dynamics of open two-state quantum systems defined over the real numbers. Its aim is twofold. On the one hand, we provide a complete mathematical description of such dynamics when the affine component of the quantum channel semigroup is Markovian. We describe Lindblad and Kraus operators, orthogonal evolutions by means of Jordan algebra derivations, as well as solutions of the GKSL dissipative equation and their effect on the relative entropy. On the other hand, we are interested in an original application to color perception that motivates this analysis. Using a recent novel model in which perceived colors are given by Lüders measurements on the state space of a rebit, we show how the chromatic distortion induced by a non-neutral illuminant can be modeled by a Markovian rebit channel that progressively reduces color distinguishability. This colorimetric effect is illustrated through simulations on digital images, offering a concrete demonstration of the relevance of Markovian rebit evolution for modeling perceptual phenomena.
This paper investigates the Markovian dynamics of open two-state quantum systems defined over the real numbers (rebits). Two main objectives are pursued. First, we present a comprehensive classification of Markovian rebit quantum channels, i.e. one-parameter semigroups of completely positive, trace-preserving (CPTP) maps acting on the rebit state space. We show that a full characterisation of their action can be achieved and that describing these channels as solutions of the GKSL equation allows us to explicitly identify the associated Lindblad generators and conditions for complete positivity. Second, we present an original application of this classification to colour perception. Using a recent model in which perceived colours arise from Lüders measurements on the rebit state space, we show how chromatic distortion induced by a non-neutral illuminant can be modelled by a Markovian rebit channel that progressively diminishes colour distinguishability. Other types of channels could be used to study colour vision deficiencies. These phenomena are illustrated by simulations on digital images, highlighting the relevance of rebit Markovian dynamics in modelling colour vision.
This paper investigates the Markovian dynamics of open two-state quantum systems defined over the real numbers (rebits). Two main objectives are pursued. First, we present a comprehensive classification of Markovian rebit quantum channels, i.e. one-parameter semigroups of completely positive, trace-preserving (CPTP) maps acting on the rebit state space. We show that a full characterisation of their action can be achieved and that describing these channels as solutions of the GKSL equation allows us to explicitly identify the associated Lindblad generators and conditions for complete positivity. Second, we present an original application of this classification to colour perception. Using a recent model in which perceived colours arise from Lüders measurements on the rebit state space, we show how chromatic distortion induced by a non-neutral illuminant can be modelled by a Markovian rebit channel that progressively diminishes colour distinguishability. Other types of channels could be used to study colour vision deficiencies. These phenomena are illustrated by simulations on digital images, highlighting the relevance of rebit Markovian dynamics in modelling colour vision.
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.
Abstract The classification of qubit channels is known since 2002. However, that of rebit channels has never been studied so far, maybe because of the scarcity of concrete rebit examples. In this paper we point out that the strategy used to classify qubit channels cannot be pursued in the rebit case and we propose an alternative which allows us to complete the rebit channel classification. This mathematical result has not only a purely abstract interest: as we shall briefly mention, it may have applications in the analysis of local properties and temporal evolution of real quantum systems and also in a recent color vision model based on quantum information.
Abstract In a very recent paper, the complete classification of rebit channels has been achieved. Here, we exploit that result to show how the classification can be used to perform an exhaustive analysis of a novel class of rebit channels induced by effects called n-Lüders channels. In the final part of the paper we propose a concrete application of these channels within the mathematical framework of a quantum theory of color perception that originated the interest about a rebit channels classification.
These notes are a short introduction to the mathematical theory of open quantum systems. They are meant to serve as an entry point into a broad research area which has applications across the quantum sciences dealing with systems subjected to external noise. The guiding idea is to let the key structures of the theory emerge from a concrete model. By working through the dissipative Jaynes-Cummings model the reader will dis- cover explicitly how irreversible dynamics arises from a unitary system-reservoir evolution. The notions of the continuous mode limit, correlation functions, spectral density appear in a natural manner and lead to the evolution equation of the open system in form of a master equation. This sets the stage for the more general analysis of completely positive, trace preserving (CPTP) maps and the study of quantum dynamical semigroups. We motivate and prove the Kraus representation theorem, the dilation theorem and the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) theorem. Working through the exercises (for which full solutions are supplied) will reinforce the ideas introduced in the main text.