Back in the 1960s, based on Wiener's thought, Shikao Ikehara (first student of N.Wiener) encouraged the progress of Hisaharu Umegaki's research from a pure mathematical aspect in order to further develop the research on mathematical methods of quantum information at Tokyo Institute of Technology. Then, in the 1970s, based on the results accomplished by Umegaki Group, Ikehara instructed the author to develop and spread quantum information science as the global information science. While Umegaki Group's results have been evaluated as major achievements in pure mathematics, their contributions to current quantum information science have not been fully discussed. This paper gvies a survey of my talk in the memorial seminar on Ikehara, in which Ikehara and Umegaki Group's contributions to design theory of quantum communication have been introduced with specific examples such as quantum relative entropy and quantum Pinsker inequality.
Back in the 1960s, based on Wiener's thought, Shikao Ikehara (first student of N.Wiener) encouraged the progress of Hisaharu Umegaki's research from a pure mathematical aspect in order to further develop the research on mathematical methods of quantum information at Tokyo Institute of Technology. Then, in the 1970s, based on the results accomplished by Umegaki Group, Ikehara instructed the author to develop and spread quantum information science as the global information science. While Umegaki Group's results have been evaluated as major achievements in pure mathematics, their contributions to current quantum information science have not been fully discussed. This paper gvies a survey of my talk in the memorial seminar on Ikehara, in which Ikehara and Umegaki Group's contributions to design theory of quantum communication have been introduced with specific examples such as quantum relative entropy and quantum Pinsker inequality.
The application of quantum mechanics to information related fields such as communication, computation and cryptography is a fast growing line of research that has been witnessing an outburst of theoretical and experimental results, with possible practical applications. On the one hand, quantum cryptography with its impact on secrecy of transmission is having its first important actual implementations; on the other hand, the recent advances in quantum optics, ion trapping, BEC manipulation, spin and quantum dot technologies allow us to put to direct test a great deal of theoretical ideas and results. These achievements have stimulated a reborn interest in various aspects of quantum mechanics, creating a unique interplay between physics, both theoretical and experimental, mathematics, information theory and computer science. In view of all these developments, it appeared timely to organize a meeting where graduate students and young researchers could be exposed to the fundamentals of the theory, while senior experts could exchange their latest results. The activity was structured as a school followed by a workshop, and took place at The Abdus Salam International Center for Theoretical Physics (ICTP) and The International School for Advanced Studies (SISSA) in Trieste, Italy, from 12–23 June 2006. The meeting was part of the activity of the Joint European Master Curriculum Development Programme in Quantum Information, Communication, Cryptography and Computation, involving the Universities of Cergy–Pontoise (France), Chania (Greece), Leuven (Belgium), Rennes1 (France) and Trieste (Italy). This special issue of Journal of Physics A: Mathematical and Theoretical collects 22 contributions from well known experts who took part in the workshop. They summarize the present day status of the research in the manifold aspects of quantum information. The issue is opened by two review articles, the first by G Adesso and F Illuminati discussing entanglement in continuous variable systems, the second by T Prosen, discussing chaos and complexity in quantum systems. Both topics have theoretical as well as experimental relevance and are likely to witness a fast growing development in the near future. The remaining contributions present more specific and very recent results. They involve the study of the structure of quantum states and their estimation (B Baumgartner et al , C King et al , S Olivares et al , D Petz et al and W van Dam et al ), of entanglement generation and its quantification (G Brida et al , F Ciccarello et al , G Costantini et al , O Romero-Isart et al , D Rossini et al , A Serafini et al and D Vitali et al ), of randomness related effects on entanglement behaviour (I Akhalwaya et al , O Dahlsten et al and L Viola et al ), and of abstract and applied aspects of quantum computation and communication (K Audenart, G M D'Ariano et al , N Datta et al , L C Kwek et al and M Nathanson et al ). We would like to express our gratitude to the European Commission, the Abdus Salam ICTP, SISSA and Eurotech SpA (Amaro, Udine, Italy) for financial and/or logistic support. Special thanks also go to the workshop secretary Marina De Comelli, and the secretaries of the Department of Theoretical Physics, University of Trieste, Sabrina Gaspardis and Rosita Glavina for their precious help and assistance.
Quantum entropy is a fundamental concept for quantum information recently developed in various directions. We will review the mathematical aspects of quantum entropy (entropies) and discuss some applications to quantum communication, statistical physics. All topics taken here are somehow related to the quantum entropy that the present authors have been studied. Many other fields recently developed in quantum information theory, such as quantum algorithm, quantum teleportation, quantum cryptography, etc., are totally discussed in the book (reference number 60).
In several complicated systems, it is important to study (1) the dynamics of state change and (2) the complexity of states of systems. Information Dynamics (ID) introduced by Ohya is a general idea constructing a theory by synthesizing the research schemes of (1) and (2) under a frame of the ID. In ID, there are two types of complexities, that is, (a) a complexity of state describing system itself and (b) a transmitted complexity between two systems. Entropies of classical and quantum information theory are the example of the complexities of (a) and (b). In quantum information theory, Ohya introduced a compound state and defined Ohya mutual entropy based on quantum relative entropy of Umegaki in 1983, he extended it to general quantum systems by using the relative entropy of Araki and Uhlmann. One can discuss the coding theorems by means of the mean entropy and the mean mutual entropy defined by the dynamical entropy. In this paper, we will discuss about complexity of quantum communication processes including quantum correlations.
Pinsker's inequality sets a lower bound on the Umegaki divergence of two quantum states in terms of their trace distance. In this work, we formulate corresponding estimates for a variety of quantum and classical divergences including f-divergences like Hellinger and χ2-divergences as well as Rényi divergences and special cases thereof like the Umegaki divergence, collision divergence, max divergence. We further provide a strategy on how to adapt these bounds to smoothed divergences.
Quantum information technologies are poised to fundamentally transform global communication. Unlike classical systems, quantum communication leverages phenomena like quantum entanglement to offer unprecedented capabilities. This not only revolutionizes security and computational methods, but also drives advancements in specialized applications such as advanced sensing and precise time synchronization. These developments hold significant implications for critical sectors including medicine and health, materials science, and energy. At the core of this transformation is the concept of a quantum Internet, built upon the unique properties of quantum entanglement. The strategic use of entanglement as a communication resource is expected to profoundly impact the future evolution of the Internet and the development of 6G networks. This paper explores the path from current quantum entanglement research to a global quantum network. It highlights the advantages of satellite-based quantum communications for routing quantum information and discusses the simulation tools essential for designing and optimizing these advanced communication systems.
The quantum entropy was introduced by von Neumann around 1932, which describes the amont of information of the quantum state itself. It was extended by Ohya for C*‐systems. The quantum relative entropy was first defined by Umegaki for σ ‐finite von Neumann algebras, which denotes a certain difference between two states. It was extended by Araki and Uhlmann for general von Neumann algebras and *‐algebras, respectively. The development of communication theory is closely connected with study of entropy theory. In order to discuss the commuinication processes, we need at least two systems, such as input and output systems. The signal of the input system is carried through a physical device, which is called a channel. The mathematical representation of the channel is a mapping from the input state space to the output state space. The mutual entropy (information) defined by using the relative entropy between the product state consist of the input state and the output state and the compound state given by the communication process, expresses the amount of information correctly transmitted through the channel from the input system to the output system. In classical communication theory, the mutual entropy was formulated by using the joint probability distribution between the input system and the output system. The (semi‐classical) mutual entropies for classical input and quantum output were defined by several researchers. In fully quantum system, there does not exist the joint probability distribution in general. Instead of the joint probability distribution, Ohya invented the quantum (Ohya) compound state, and he introduced the fully quantum mutual entropy (information), which is called Ohya mutual entropy, for quantum input and output systems, describes the amount of information correctly sent from the quantum input system to the quantum output system through the quantum channel. It was extended by Ohya for general quantum systems. Recently Shor and Bennet et al took the coherent entropy and defined the mutual type entropy to discuss a sort of coding theorem for communication processes.In this paper, we concretely calculate these mutual entropy‐type measures for the attenuation channel in order to assay these abilities.