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Application of Quantum Pinsker Inequality to Quantum Communications ISSN 2186-6570 Application of quantum Pinsker inequality to quantum communications Osamu Hirota Research Center for Quantum Communication, Quantum ICT Research Institute, Tamagawa University 6-1-1 Tamagawa-gakuen, Machida, Tokyo 194-8610, Japan Tamagawa University Quantum ICT Research Institute Bulletin, Vol.9, No.1, 1-7, 2019 ©Tamagawa University Quantum ICT Research Institute 2019 All rights reserved. No part of this publication may be reproduced in any form or by any means electrically, mechanically, by photocopying or otherwise, without prior permission of the copy right owner. Tamagawa University Quantum ICT Research Institute Bulletin 1 Vol.9 No.1 : 1―7（2019） Application of quantum Pinsker inequality to quantum communications Osamu Hirota Research Center for Quantum Communication, Quantum ICT Research Institute, Tamagawa University 6-1-1 Tamagawa-gakuen, Machida, Tokyo 194-8610, Japan E-mail: [email protected] Abstract—Back in the 1960s, based on Wiener’s thought, clearly defines very common signals processed in human Shikao Ikehara (first student of N.Wiener) encouraged the social activities and it is applied to the communication progress of Hisaharu Umegaki’s research from a pure system with the operational meanings for the relevant mathematical aspect in order to further develop the research on mathematical methods of quantum information at Tokyo information processing. Quantum information theory as Institute of Technology. Then, in the 1970s, based on the mathematics can be modeled on it. However, unlike Shan- results accomplished by Umegaki Group, Ikehara instructed non information, quantum entropy lacks versatility regard the author to develop and spread quantum information for operational meaning on information transmission. It is science as technology. While Umegaki Group’s results have defined in consideration of mathematical form or applica- been evaluated as major achievements in pure mathematics, their contributions to current quantum information science tion to physics. Therefore, no new information scientific have not been fully discussed. This paper will clarify technology is expected from simple generalization of Umegaki Group’s contributions to design theory of quantum the concepts of Wiener and Shannon. Only when its communication with specific examples. contribution to Wiener-Shannon system is proved, the mathematics is deemed to have contributed to information I. INTRODUCTION science. In the real world, we have no performance evaluation It was Holevo who discovered a liaison between quan- measures for communication system with operational tum entropy and Shannon information. In the study of meanings of information transmission and processing accessible information ( an application of Shannon’s other than various signal detection criteria established by mutual information) to quantum system, its upper bound Wiener, and Shannon entropy by Shannon. The same is is now called Holevo bound, which is given in the form true even where the physical system for implementation of quantum entropy, and its formula is called Holevo is generalized into quantum system or relativistic system, information. Holevo derived the upper bound in flow of which means that modern communication theory is the the result of Stratonovich on Nth extended system in most successful field among other scientific theories in relation to Shannon mutual information in quantum sys- human history. tem (see Reference 8). Prior to that, Tamagawa University Quantum information science originates from quantum Group had published a paper providing specific examples communication theory, based on which, quantum commu- of maximization of accessible information and super- nication such as quantum key distribution and quantum additivity in Nth extended systems. Jozsa Group proved symmetric key cipher has been developed. These are that the maximum amount of accessible information formulated based on and as quantum versions of statis- reaches Holevo information in the limit of pure quantum tical signal detection theory and Shannon’s information state system without external noise. Jozsa presented the transmission theory. The former has a beautiful form as results at the 3rd International Conference on Quan- a design theory for detection and estimation techniques tum Communication, Measurement (QCM 1996). While of signals transmitted in a quantum state established by Holevo had shifted to research on quantum stochastic Helstrom [1], Holevo [2] and Yuen [3]. The theory of process, I recommended him that he should generalize Shannon information transmitted in a quantum state was Jozsa’s works. In a very short while, Holevo and Tam- started by Stratonovich, Holevo, et al and studied by agawa University Group discussed the proof method in Yuen [4] and Hirota [5] from the viewpoint of quantum the general model of external noise system with Kitaev, state control as well as by Jozsa Group [6] and Masashi and only after a week later, Holevo showed us the Ban of Tamagawa University Group [7] in the context of proposed proof on a white board at Tamagawa University. accessible information. We conveyed the results to Jozsa and Holevo published On the other hand, in mathematics, theories may be his proof in IEEE’s Transaction on Information Theory. developed without regard for the operability of infor- Jozsa Group also started their consideration and the mation handled by humans or with a focus on physi- proof was completed by Shumacher-Westmoreland and cal phenomena of a specific device. Shannon’s entropy published in the Physical Review. Although published at 2 different times, these are now called Holevo-Shumacher- system with the quantum density operator: Westmoreland theorem as the formula of discrete channel ρ (H ) is Xq ∈D S capacity of classical-quantum composite system [9, 10]. S(X ) S(ρ )= Tr ρ log ρ (6) Additionally, Holevo and Tamagawa University Group q ≡ Xq − { Xq Xq } derived continuous channel capacity and formula for This is called von Neumann entropy. Research on a reliability function in Gaussian System [11, 12]. As a quantum version of relative entropy, which is regarded result, Holevo information, which is expressed in the form as a mathematical generalization of Shannon’s entropy of quantum entropy and serves as a parameter of the theory, was begun for the first time in the world by extreme point of Shannon system, has greatly contributed Hisaharu Umegaki at Tokyo Institute of Technology. to the real world. This was driven by the motivation of Shikao Ikehara to On the other hand, a formal formulation of Shannon recommend the succession of Wiener thought [14,15]. theory was developed as mathematics in a quantum en- Umegaki, for the first time in the world, defined the tropy form without considering the operational meaning following quantum relative entropy on the von Neumann like Shannon information. The question in this paper is algebra in 1962 and formulated its various features [16- whether it has operational meaning in the actual commu- 18]. nication system. Definition 1(Umegaki) II. PROGRESS IN QUANTUM ENTROPY − A. Approach of Holevo and Umegaki Dq(ρ σ)=Tr ρ[log ρ log σ] (7) Let us here denote the most fundamental formula in || { − } supp(ρ) supp(σ) (8) Shannon’s communication theory in the following. ⊆ On the other hand, Helstrom and Holevo inherited the H(X)= P (x) log P (x) (1) − idea of Wiener- Shannon, and they defined that, in a x ∑ communication system using quantum phenomena, the H(X Y )= P (y)P (x y) log P (x y) (2) | − | | sender prepares a set of quantum density operators: y x x ∑ ∑ ϵ = p(x),ρY . That is, a message x is mapped I(X, Y )=H(X) H(X Y )=H(Y ) H(Y X) { q } − | − | to a quantum density operator which corresponds to a = H(X)+H(Y ) H(X, Y ) (3) concrete signal. The quantum density operator of the set − is described as follows: Shannon’s information H(X) and mutual information x I(X, Y ) have a clear operational meaning according to ρYq = p(x)ρYq (9) x the coding theorem. In physics, mutual information is ∑ sometimes regarded as a measure of classical correlation The receiving system performs a quantum measurement between two statistical systems. Meanwhile, entropy was on the quantum system, and becomes a model for deter- introduced in 1951 by Kullback in statistics [13]. In that mining the classical parameter x as a message. This { } case, there is a strong intention to express the distance is a problem of Accessible information (Shannon mutual between probability distributions representing the charac- information of classical-quantum composite systems). teristics of two statistical systems. Therefore, they start based on the following relative entropy: Iacc = max I(Xc,Yq) (10) Π P (x) D (P (x) Q(x)) = P (x) log (4) where Π is detection operator or positive operator valued c || Q(x) x measure (POVM). That is, to preserve Shannon’s view ∑ One can generalize the above into the composite sys- of the world, Holevo considers the set whose signal tem. If we denote the joint probability on the composite element is the classical parameter of the quantum density operator, ϵ = p(x),ρx . And the quantum entropy of system as follows: P (x, y),Q(x, y)=P (x)P (y) we { Yq } have the classical-quantum composite system was defined in 1973 as follows. P (x, y) D (P (x, y) Q(x, y)) = P (x, y) log c || P (x)P (y) x y Definition 2(Holevo) ∑ ∑ − = I(X, Y ) (5) χ(ϵ)=S(ρ ) p(x)S(ρx ) (11) Yq − Yq x Thus, from a mathematical definition point of view, the ∑ relative entropy looks like “General”. This is called the Holevo information. As stated in On the other hand, von Neumann defined entropy the introduction, the Holevo -Shumacher-Westmoreland for quantum systems in response to the development of theorem guarantees that the limit of Shannon information quantum statistical mechanics. The entropy of a quantum transmitted in a quantum system is the maximum value 3 different times, these are now called Holevo-Shumacher- system with the quantum density operator: of Holevo information.
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