In information communication theory, one can study the efficiency of information transmission of the communication processes by using the Shannon's type inequalities by means of the entropy and the mutual entropy. In this paper, we discuss about entropy type inequalities for treating the classical Gaussian communication processes consistently.
In information communication theory, one can study the efficiency of information transmission of the communication processes by using the Shannon's type inequalities by means of the entropy and the mutual entropy. In this paper, we discuss about entropy type inequalities for treating the classical Gaussian communication processes consistently.
In order to discuss the efficiency of information transmission of the Gaussian communication processes consistently, we introduced an entropy type functional and a mutual entropy type functional in reference [16]. In that study, we used the set of Gaussian input measures with covariance operators of trace one. In this paper, we refine the formulation and modify the entropy type complexity and the mutual entropy type complexity to be able to study the efficiency of information transmission for more general input Gaussian measures and Gaussian channels.
In classical information theory, one of the most important theorems are the coding theorems, which were discussed by calculating the mean entropy and the mean mutual entropy defined by the classical dynamical entropy (Kolmogorov-Sinai). The quantum dynamical entropy was first studied by Emch [13] and Connes-Stormer [11]. After that, several approaches for introducing the quantum dynamical entropy are done [10, 3, 8, 39, 15, 44, 9, 27, 28, 2, 19, 45]. The efficiency of information transmission for the quantum processes is investigated by using the von Neumann entropy [22] and the Ohya mutual entropy [24]. These entropies were extended to S- mixing entropy by Ohya [26, 27] in general quantum systems. The mean entropy and the mean mutual entropy for the quantum dynamical systems were introduced based on the S- mixing entropy. In this paper, we discuss the efficiency of information transmission to calculate the mean mutual entropy with respect to the modulated initial states and the connected channel for the quantum dynamical systems.
The classical entropies were formulated by means of the classical probability theory. The essence of the classical probability theory corresponds to the use of c‐number with commutative property. On the base of the classical probability theory, the classical information theory can be discussed by using the classical entropies. On the other hand, the principle of the quantum phenomenon seems to be having non‐commutative behavior. It leads to the use of the q‐operator with non‐commutative property to introduce the quantum entropies based on the quantum probability theory. In order to discuss the efficiency of the information communication processes, a measure of complexity of initial state itself and a measure of transmitted complexity through communication channels are necessary. Shannon’s entropy and the mutual entropy represent these measures of complexities in classical communication system, respectively. Von Neumann entropy and Ohya mutual entropy relate to these measures of complexities in quantum communication system, respectively. Recently, several mutual entropy type measures (Lindblad—Nielsen entropy and Coherent entropy) were proposed by using an entropy exchange with respect to a channel and an initial state. In this paper, we will show the most suitable one for discuss the efficiency of information transmission for quantum processes. In classical information theory, one of the most important tools to analyze communication processes are the classical coding theorems, which was investigated through the classical dynamical entropy. The classical dynamical entropy was discussed in [1] with respect to a measure preserving transformation defined on a message space for finite partitions of the measure space. In quantum information theory, Emch [2] and Conne, Stormer [3] first studied a complexity of quantum dynamical processes. After that, Ohya introduced a S‐ mixing entropy in general quantum systems, which was natural extension of the von Neumann entropy. Using the S—mixing entropy, Ohya defined a mean entropy and a mean mutual entropy for the quantum dynamical systems. Conne, Narnhoffer and Thirring introduced a dynamical entropy (CNT entropy) [4] and several researchers discussed the CNT entropy [5,6,7,8]. Alicki and Fannes defined a dynamical entropy (AF entropy) [9] using a finite operational partitions of unity. In 1995, Voiculescu proposed a dynamical entropy (dynamical approximation entropy) [10] in C* and W* algebras. Through the quantum Markov process, Accardi, Ohya and Watanabe defined a dynamical entropy (AOW entropy) [11,12] in 1997. Base on transition expectations and liftings in the sense of Accardi and Ohya [13], Kossakowski, Ohya and Watanabe introduced a dynamical entropy (KOW entropy) [14] with respect to completely positive maps in 1999. In this paper, we will discuss the complexity of the quantum dynamical process to calculate the generalized AOW entropy given by KOW entropy.
This paper introduces a novel approach to information theory, leveraging quantum entanglement entropy as a fundamental measure of information. We propose that entanglement entropy can effectively quantify the efficiency of information transmission and provides a theoretical foundation that surpasses the limitations of classical information theory. The core of this work lies in the integration of entanglement entropy with Shannon entropy, establishing a new framework for analyzing information transfer. We explore scenarios where classical information theory struggles to adequately describe the limits of communication due to quantum mechanical effects, specifically entanglement. The paper aims to provide a theoretical underpinning for quantum communication protocols and potentially inform the development of quantum computation. We define key terms and formulate core concepts, focusing on the relationship between entanglement entropy, information content, and the constraints imposed by quantum mechanics. The central argument is that by considering the inherent interconnectedness of entangled systems, we can gain a deeper understanding of information processing at the most fundamental level. ---
The quantum entropy introduced by von Neumann around 1932 describes the amount of information of the quantum state itself. It was extended by Ohya for C*-systems before Conne-Narnhoffer-Thirring (CNT) entropy. The quantum relative entropy was first defined by Umegaki for σ-finite von Neumann algebras and it was subsequently extended by Araki and Uhlmann for general von Neumann algebras and *-algebras, respectively. By introducing a new notion, the so-called compound state, in 1983 Ohya succeeded to construct the mutual entropy in a complete quantum mechanical system (i.e., input state, output state and channel are all quantum mechanical) describing the amount of information correctly transmitted through the quantum channel. In this paper, we briefly review Ohya's S-mixing entropy and the quantum mutual entropy for general quantum systems. Based on a concept of structure equivalent, we apply the general framework of quantum communication to the Gaussian communication processes.
Abstract One of most important issues in quantum information theory concerns transmission of information through noisy quantum channels. We discuss a few channel characteristics expressed by means of generalized entropies. Such characteristics can often be treated in line with more usual treatment based on the von Neumann entropies. For any channel, we show that the q-average output entropy of degree q ≥ 1 is bounded from above by the q-entropy of the input density matrix. The concavity properties of the (q, s)-entropy exchange are considered. Fano type quantum bounds on the (q, s)-entropy exchange are derived. We also give upper bounds on the map (q, s)-entropies in terms of the output entropy, corresponding to the completely mixed input.