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arXiv:2106.14228

Majorization effect on entropic functionals: An application to a V-type three-level atom-field interacting system

Debraj Nath

Published Jun 27, 2021• 0 citations
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Abstract

Majorization effect on some entropic functionals, such as von-Neumann, Shannon, atomic Wehrl and Rényi entropies are investigated of a V-type three-level atom, which interacts with a coherent field in a resonant cavity. Fidelity, purity and linear entropy are investigated for two quantum systems, which contain photon number distributions and Husimi Q functions. A relation between majorization and localization properties of continuous density functionals is established. The results are compared and verified for continuous and discrete distributions of an atom-field interacting system.

DOI
10.48550/arxiv.2106.14228

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Majorization effect on some entropic functionals, such as von-Neumann, Shannon, atomic Wehrl and Rényi entropies are investigated of a V-type three-level atom, which interacts with a coherent field in a resonant cavity. Fidelity, purity and linear entropy are investigated for two quantum systems, which contain photon number distributions and Husimi Q functions. A relation between majorization and localization properties of continuous density functionals is established. The results are compared and verified for continuous and discrete distributions of an atom-field interacting system.
The majorization effect on some entropic functionals, such as von-Neumann, Shannon, atomic Wehrl and Renyi entropies are investigated of a vee type three-level atom which interacts with a coherent field in a resonant cavity. Moreover, the fidelity, purity and linear entropy are investigated of two quantum systems, which contain photon number distributions (discrete) and Husimi QQQ function (continuous distribution). A relation between the majorization and localization properties of continuous density functionals is established. The results are compared and verified for continuous and discrete distributions.
The theory of majorization is a powerful mathematical tool which naturally arises in the quantum theory as a consequence of its fundamental connection with unitarity. In this work, we look for majorization relations based on entropic inequalities. We begin by introducing the theory of majorization. We then give some notions of quantum mechanics, like the density matrix and some algebra, before presenting the concept of entropy which is closely related to the theory of majorization. We define in the process the Shannon, von Neumann and Renyi entropies, along with some of their properties and applications. We then define quantum entanglement and give some criteria for separability, which turn out to be applications of the concepts of majorization and entropy in the quantum theory. We present some new majorization relations similar to already existing entropic inequalities, and exhibit their advantages and limitations. The rest of the report is dedicated to the study of the field of Gaussian quantum information. We present the theory, before introducing other majorization relations, which turn out to be useful because they allow one to know if it is possible to transform one quantum state into another when there exists majorization relations between them.
We present a continuity bound for quantum conditional entropies in terms of fidelity, with no dependence on the dimension of the conditioning system. Our results encompass both the von Neumann entropy and one version of sandwiched Rényi entropy, which is particularly relevant for applications in quantum key distribution (QKD). In such applications, this bound should provide improved results as compared to previous continuity bounds based on trace distance. We also show that the bounds can be sharpened as expected for classical-quantum states, and verify that they hold when the conditioning system has countably infinite dimension. Finally, we present several variants tailored for exponential functions of the Rényi entropies instead, which should be relevant for QKD applications.
We review the concept of majorization and its relation with generalized information measures. Majorization theory provides an elegant framework for comparing two probability distributions, leading to a rigorous concept of disorder which is more stringent than that based on the Shannon entropy. Nevertheless, it is shown that it can be fully captured through general entropic inequalities based on generalized entropic forms. A brief review of generalized entropies is also provided. As illustration, we discuss the majorization properties of generalized thermal distributions derived from generalized entropies, and identify rigorous mixing parameters. We also describe majorization in quantum systems. We discuss in particular its capability for providing a disorder based criterion for the detection of quantum entanglement, which is stronger than that based on the von Neumann entropy and leads to a generalized entropic separability criterion.
In this paper, the Wehrl entropy approach is discussed and compared with the quantum entanglement using a mixed-state three-level atom interacting with a cavity field. In the pure state case, the behavior of the atomic Wehrl entropy shows the same behavior of the entanglement due to the von-Neumann entropy, while the mixed state case gives the total correlation due to quantum mutual entropy. If the system is in an entangled state, the formalism can be used to quantify the entanglement as well as the total correlations.