Fidelity measure and conservation of information in general probabilistic theories

Abstract

-We investigate the main features of a measure of fidelity between states in a general family of probabilistic theories admitting classical probability theory and standard quantum theory as particular instances. We apply the aforementioned measure to investigate informationtheoretical features of these theories related to the conservation of information during the evolution of closed physical systems. In particular, we derive a generalization of a fundamental result in quantum theory relevant for the measurement problem: Zurek's recent extension of the no-cloning theorem. Copyright c EPLA, 2009Introduction. -The physics of information and computation [1][2][3][4][5][6] has been the focus of an intense and increasing research activity in recent years [5][6][7][8][9][10][11][12][13]. Part of this research effort has been devoted to determine the ultimate limits imposed by the fundamental laws of physics on any device processing or transmitting information [6][7][8].On the other hand, a growing body of theoretical developments indicate that the concept of information constitutes an essential ingredient for a deep understanding of physical systems and processes [1][2][3][4][5][6][9][10][11][12]. Ideas, techniques, and models inspired by the theoretical analysis of information processing devices proved to be relevant for the study of a variegated range of physical scenarios, including applications to subjects as diverse as quantum thermodynamical machines [9] or the fundamental limits on the accuracy of spacetime measurements [10]. Interesting attempts have been recently made to derive the basic formalism of quantum theory from information-theoretical concepts [11]. The advent of quantum information science, and the concomitant discovery of the novel, subtle, and counter-intuitive ways of processing and transmitting information allowed by quantum mechanics [6,13] greatly stimulated the interest in these lines of enquiry. Work on quantum information

Classical analogue of the statistical operator

“…Therefore, we have (27) which illustrates the fact that D A constitutes a measure of statistical correlations between subsystems A and B. Indeed, one can see in this example that D A is a monotonically increasing function of the mutual information I AB (see Figure 1).…”

“…This kind of dynamics constitutes (at the classical level) the most fundamental one. According to our present understanding of Nature, the basic laws of physics satisfy this information-preserving property [26,27]. Indeed, the conservation of information has been hailed by some leading theoreticians as the most fundamental law of physics [28].…”

Measurement interpretation and information measures in general probabilistic theory

“…Zander and Plastino investigate the main features of a measure of fidelity between states in GPT, and use this quantity to study information-theoretical features of GPT related to conservation of information during the evolution of closed physical systems. Particularly, they derived a generalization of Zurek's result under the framework of GPT [16]. In this paper, we reconsider Luo's derivation in the GPT regime.…”

“…[2,9,16]), which was also called BBLW framework in [16]. Note that the mathematical structure is physically motivated.…”

“…The properties of the quantity in Eq. (2) are extensively studied in [9] and [16]. Specifically, the following lemma is crucial in proceeding our derivation.…”