Selected Papers of M Ohya

The following sections are included:

• Preface

• Contents

• Introduction

• Curriculum Vitae

I will discuss the following four (1)-(4) below from both mathematical and philosophical views: (1) What is (or do we mean) the understanding of the existence ? (2) We propose “Adaptive dynamics” to understand the existence. (3) The adaptive dynamics can be used to describe chaos. (4) The adaptive dynamics is applied to the SAT Quantum Algorithm to solve the NP complete problem.

The success of quantum mechanics is due to the discovery that nature is described in infinite dimension Hilbert spaces, so that it is desirable to demonstrate the quantum teleportation process in a certain infinite dimensional Hilbert space. We describe the teleportation process in an infinite dimensional Hilbert space by giving simple examples.

It is von Neumann who opened the window for today's information epoch. He defined quantum entropy including Shannon's information more than 20 years ahead of Shannon, and he explained what computation means mathematically.

In this paper I discuss two problems studied recently by me and my coworkers. One of them concerns a quantum algorithm in a generalized sense solving the SAT problem (one of NP complete problems) and another concerns quantum mutual entropy properly describing quantum communication processes.

There exists an important problem whether there exists an algorithm to solve an NP-complete problem in polynomial time. In this paper, a new concept of quantum adaptive stochastic systems is proposed, and it is shown that it can be used to solve the problem above.

ordinary approach to quantum algorithm is based on quantum Turing machine or quantum circuits. It is known that this approach is not powerful enough to solve NP-complete problems. In this paper we study a new approach to quantum algorithm which is a combination of the ordinary quantum algorithm with a chaotic dynamical system. We consider the satisfiability problem as an example of NP-complete problems and argue that the problem, in principle, can be solved in polynomial time by using our new quantum algorithm.

We propose a new approach to define chaos in dynamical systems from the point of view of Information Dynamics. Observation of chaos in reality depends upon how to observe it, for instance, how to take the scale in space and time. Therefore it is natural to abandon taking several mathematical limiting procedures. We take account of them, and chaos degree previously introduced is redefined in this paper.

The quantum capacity of a pure quantum channel and that of classical-quantum-classical channel are discussed in detail based on the fully quantum mechanical mutual entropy. It is proved that the quantum capacity generalizes the so-called Holevo bound.

The operational structure of quantum couplings and entanglements is studied and classified for semi-finite von Neumann algebras. We show that the classical–quantum correspondences, such as quantum encodings, can be treated as diagonal semi-classical (d-) couplings, and the entanglements, characterized by truly quantum (q-) couplings, can be regarded as truly quantum encodings. The relative entropy of the d-compound and entangled states leads to two different types of entropy for a given quantum state: the von Neumann entropy, which is achieved as the maximum of mutual information over all d-entanglements, and the dimensional entropy, which is achieved at the standard entanglement (true quantum entanglement) coinciding with a d-entanglement only in the case of pure marginal states. The d-and q-information of a quantum noisy channel are, respectively, defined via the input d- and q-encodings, and the q-capacity of a quantum noiseless channel is found to be the logarithm of the dimensionality of the input algebra. The quantum capacity may double the classical capacity, achieved as the supremum over all d-couplings (or encodings) bounded by the logarithm of the dimensionality of a maximal Abelian subalgebra.

We study the chaotic behavior and the quantum-classical correspondence for the Baker's map. Correspondence between quantum and classical expectation values is investigated and it is numerically shown that it is lost at the logarithmic timescale. The quantum chaos degree is computed and it is demonstrated that it describes the chaotic features of the model. The correspondence between classical and quantum chaos degrees is considered.

Following the previous paper in which quantum teleportation is rigorously discussed with coherent entangled states given by beam splittings, we further discuss two types of models, the perfect teleportation model and non-perfect teleportation model, in a general scheme. Then the difference among several models, i.e., the perfect models and the non-perfect models, is studied. Our teleportation models are constructed by means of coherent states in some Fock space with counting measures, so that our model can be treated in the frame of usual optical communication.

Quantum teleportation is rigorously demonstrated with coherent entangled states given by beam splittings. The mathematical scheme of beam splitting has been used to study quantum communication [2] and quantum stochastic [8]. We discuss the teleportation process by means of coherent states in this scheme for the following two cases: (1) Delete the vacuum part from coherent states, whose compensation provides us a perfect teleportation from Alice to Bob. (2) Use fully realistic (physical) coherent states, which gives a non-perfect teleportation but shows that it is exact when the average energy (density) of the coherent vectors goes to infinity. We show that our quantum teleportation scheme with coherent entangled state is more stable than that with the EPR pairs which was previously discussed.

In complexity theory, a famous unsolved problem is whether NP is equal to P or not. In this paper, we discuss this aspect in SAT (satisfiability) problem, and it is shown that SAT can be solved in polynomial time by means of a quantum algorithm if the superposition of two orthogonal vectors |0〉 and |1〉 prepared is detected physically.

Chaos degree defined through two complexities in information dynamics is applied to some deterministic dynamical models. It is shown that this degree well describes the chaotic feature of the models.

The rigorous formulation of the quantum mutual entropy is reviewed. The capacities for various channels, such as quantum channel, classical-quantum channel and classical-quantum-classical channel are discussed exhaustively and some misuses of these capacities are indicated.

In Section 1 we introduce the notion of lifting as a generalization of the notion of compound state introduced in [21] and [22] and we show that this notion allows a unified approach to the problems of quantum measurement and of signal transmission through quantum channels. The dual of a linear lifting is a transition expectation in the sense of [3] and we characterize those transition expectations which arise from compound states in the sense of [22].

In Section 2 we characterize those liftings whose range is contained in the closed convex hull of product states and we prove that the corresponding quantum Markov chains [2] are uniquely determined by a classical generalization of both the quantum random walks of [4] and the locally diagonalizable states considered in [3].

In Section 4, as a first application of the above results, we prove that the attenuation (beam splitting) process for optical communication treated in [21] can be described in a simpler and more general way in terms of liftings and of transition expectations. The error probabilty of information transmission in the attenuation process is rederived from our new description. We also obtain some new results concerning the explicit computation of error probabilities in the squeezing case.

A dynamical entropy for not only shift but also completely positive (CP) map is defined by generalizing the AOW entropy¹ defined through quantum Markov chain and AF entropy defined by a finite operational partition. Our dynamical entropy is numerically computed for several models.

The concept of complexity in Information Dynamics is discussed. The chaos degree defined by the complexities is applied to examine chaotic behavior of logistic map.

We analyze the variation of HIV after infection by means of an information measure, called the entropy evolution rate. In our analysis, we use a part of the external glycoprotein gp120 including the V3 region observed from six patients.

Then we could make the following two aspects clear;

(1) the relation between the change of the entropy evolution rate and the appearance of symptoms of disease, and

(2) the relation between the change of the entropy evolution rate and that of the CD4 count of the patients.

Capacities of quantum mechanical channels are defined in terms of mutual information quantities. Geometry of the relative entropy is used to express capacity as a divergence radius. The symmetric quantum spin 1/2 channel and the attenuation channel of Boson fields are discussed as examples.

The complexities in information dynamics are reviewed and their examples are given. The fractal dimensions of a quantum state are discussed from a general point of view of complexity. It is shown trough a model that the fractal dimensions of a state provide measures for order structure of chaotic systems.

The present paper consists of two parts. In the first one it will be proved that the von Neumann entropy governs the size of rather sure projections in the course of independent trials. The second part is devoted to the extension of the von Neumann entropy to states of arbitrary unital C^*-algebras.

Classical dynamical entropy is an important tool to analyse the efficiency of information transmission in communication processes. Quantum dynamical entropy was first studied by Connes, Størmer and Emch. Since then, there have been many attempts to formulate or compute the dynamical entropy for some models. Here we review four formulations due to (a) Connes, Narnhofer and Thirring, (b) Ohya, (c) Accardi, Ohya and Watanabe, (d) Alicki and Fannes. We consider mutual relations between these formulations and we show some concrete computations for a model.

Two functionals and Ĩ are introduced for C^*-dynamical systems with invariant states and stationary channels. It is shown that the Kolmogorov–Sinai-type theorems hold for these functionals and Ĩ. Our functionals and Ĩ are set within the framework of quantum information theory and generalize a quantum KS entropy by CNT and the mutual entropy by Ohya.

Various physical or nonphysical systems can be described by states, so that the dynamics of a system is described by the state change. One of essential characters of a state is expressed by its complexity. Complexity such as entropy is a key concept in Information theory. We call the study of the state change together with such complexities “Information dynamics”, which is a kind of synthesis of dynamics of state change and information theory. Here we explain what information dynamics is and indicate how it can be used in optical communication. Some of concrete applications of information dynamics are discussed in the papers [9,10] of this volume.

Some concepts in information theory are tried to apply to the study of genes. The mutual entropy is used to define a measure indicating the similarity between two genetic sequences. The alignment of sequences is briefly discussed. Some phylogenetic trees are written by using the entropy measure. According to this results, usefulness of information theory is discussed in the study of genes such as molecular evolution.

Several quantum entropies are systematically studied and the mathematical structure of a channel in optical communication processes is presented. As applications of these entropies and channel, general formulas of error probability in some communication processes using, for instance, coherent or squeezed states, are obtained and the irreversibility for some dynamical processes is discussed.

A quantum mechanical compound state of an input state and its output state generated through a communication channel is constructed. The mutual information of quantum communication theory is defined by using the compound state, and its fundamental properties are studied.

When a state of a physical system dynamically changes to another state, it is important to know the correlation existing between the initial state and the final state. This correlation is described by a compound state (measure) in classical systems. In this note, we show a way how to construct such a compound state in quantum systems which is an extension of the classical compound state.

The quantum ergodic channel is studied by operator algebraic methods. The ergodic and KMS channels are introduced and their dynamical properties are discussed.

The sufficiency in von Neumann algebras is discussed with some applications to classification of normal states. It is shown that the concept of sufficiency characterizes the KMS-states and the invariant states with respect to a modular automorphism group. The relations between the Sufficiency and the relative entropy are established.

The open system dynamics is rigorously studied within the C^*-algebraic framework in terms of the approach to equilibrium. It is pointed out that every combined state of every state of a finite system and an equilibrium state describing an infinite reservoir relaxes to equilibrium through an interaction between both systems when the total Hamiltonian of the combined system satisfies some spectral properties.

The problems of a dynamical process in the linear response theory are studied in the C^*-algebraic framework. We outline the domain of validity of the linear response approximation by considering the similarity and difference between the linear response dynamics and the exact dynamics. It is shown that the linear response method is identical to the exact method when we consider the return to equilibrium of all locally perturbed states. As far as the stability of a dynamical system is concerned, the linear response theory is shown to differ from the exact theory. It is pointed out that the phase transition in a dynamical process does not occur in the linear response theory.

The problems of stability and approach to equilibrium of the Weiss Ising model are studied. Our investigations are performed in the exact and linear response senses in order to compare both theories. The change of a metastable state of the Weiss Ising model is discussed under local perturbations.

The following sections are included:

• List of Publications