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Communication in Statistics- Theory and Methods

Quantum entropy in terms of local quantum Bernoulli noises and related properties

Qi Han, Zhihe Chen, Ziqiang Lu

Published Aug 31, 2020• 17 citations
Publisher

Abstract

Localization of quantum Bernoulli noises (LQBNs) are the family of local annihilation and local creation operators acting on Bernoulli functionals. In this paper, we construct a density operator ρk represented by LQBNs, and define a new quantum entropy S(ρk) based on LQBNs. Furthermore, we demonstrate that this quantum entropy S(ρk) also has the basic properties of von Neumann entropy, such as concavity, subadditivity, and nonnegativity. In particular, we obtain the necessary and sufficient condition for S(ρk)=k.

DOI
10.1080/03610926.2020.1812654

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Localization of quantum Bernoulli noises (LQBNs) are the family of local annihilation and local creation operators acting on Bernoulli functionals. In this paper, we construct a density operator ρk represented by LQBNs, and define a new quantum entropy S(ρk) based on LQBNs. Furthermore, we demonstrate that this quantum entropy S(ρk) also has the basic properties of von Neumann entropy, such as concavity, subadditivity, and nonnegativity. In particular, we obtain the necessary and sufficient condition for S(ρk)=k.
Abstract In the era of rapid development of quantum mechanics, some issues in the field of quantum information have become increasingly important. Local quantum Bernoulli noises (LQBNs) is a localized Bernoulli functional, and it is the localization of quantum Bernoulli noises (QBNs). The distinctive feature of LQBNs lies in its localization properties, which make it highly valuable in the field of quantum information. In this paper, we study three new types of quantum conditional entropy with localization characteristics from three dimensions: local quantum conditional entropy, local quantum conditional entropy at the same position, and classical-quantum conditional entropy. Based on the locality of LQBNs, we prove that the local quantum conditional entropy is concave and unitary invariant. Surprisingly, we find that the local quantum mutual information (entropy) corresponding to the same position is 0. The local quantum conditional entropy defined based on the local quantum state at a specific position, proposed in this paper, provides new insights into understanding the impact of local quantum noise on the information transmission in quantum channels. This discovery not only deepens our understanding of the internal structure of local quantum entropy, but also reveals its influence on the performance of quantum communication.
Quantum noise has always been an important issue in the field of quantum information transmission. As the localization of quantum noise, local quantum Bernoulli noises (LQBNs) have attracted extensive attention in recent years. In this paper, the quantum conditional entropy based on LQBNs is deeply discussed, and a new definition of quantum conditional entropy is given. Then we find that this quantum conditional entropy has some basic properties such as concavity, conditional reduced entropy and unitary invariance. This research is of great significance to the field of quantum information, which can help us to understand the influence of local quantum noises on quantum information transmission.
Local Quantum Bernoulli Noises (LQBNs) is the localization of Quantum Bernoulli Noises (QBNs), which has high research value in the field of quantum information and has received widespread attention in recent years. In this paper, a new entropy exchange and coherent information with localization characteristics are studied. According to the quantum channel constructed by LQBNs, the relationship between coherent information and coherent mutual entropy with localization characteristics is proved. Finally, we also prove that the coherent information based on LQBNs may disappear.
In this paper we show that annihilation and creation operators on the space of Bernoulli functionals in discrete time can be naturally interpreted as discrete-time quantum Bernoulli noises (QBNs). We first examine their algebraic properties and prove among other things that the family of annihilation-creation sum operators generates a commutative group of unitary operators. Additionally, then by using QBN we prove an operator version of the discrete-time Clark formula. We also obtain a QBN-based necessary and sufficient condition for an operator process to be adapted. We define integrals of adapted operator processes with respect to QBN and show their operator martingale property. Finally, we give a representation formula for operators on the space of Bernoulli functionals.
Abstract Chapter 20 provides a discussion of entropies and entropic quantities in quantum information theory, briefly considering the generalizations of the Rényi entropy to the quantum regime, but mainly focusing on the von Neumann entropy and its properties such as concavity, the Araki-Lieb inequality, and subadditivity. We introduce the quantum relative entropy, quantum joint entropy, quantum conditional entropy, and quantum mutual information, and prove key properties and results for these quantities, including Klein’s inequality, joint convexity, additivity, and monotonicity under completely positive and trace-preserving maps of the relative entropy, as well as weak monotonicity and strong subadditivity of the von Neumann entropy. We then analyse the relation of (negative) conditional entropy and entanglement. In this context we discuss the conditional amplitude operator and the mutual amplitude operator, as well as conditional Rényi entropies and their role for entanglement detection.