Masanori Ohya

Some aspects of quantum information theory and their applications to irreversible processes

Abstract 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.

Mathematical Foundations of Quantum Information and Computation and Its Applications to Nano- and Bio-systems

This monograph provides a mathematical foundation to the theory of quantum information and computation, with applications to various open systems including nano and bio systems. It includes introductory material on algorithm, functional analysis, probability theory, information theory, quantum mechanics and quantum field theory. Apart from standard material on quantum information like quantum algorithm and teleportation, the authors discuss findings on the theory of entropy in C*-dynamical systems, space-time dependence of quantum entangled states, entangling operators, adaptive dynamics, relativistic quantum information, and a new paradigm for quantum computation beyond the usual quantum Turing machine. Also, some important applications of information theory to genetics and life sciences, as well as recent experimental and theoretical discoveries in quantum photosynthesis are described.

COMPLEXITIES AND THEIR APPLICATIONS TO CHARACTERIZATION OF CHAOS

The concept of complexity in InformationDynamics is discussed. The chaos degree defined by thecomplexities is applied to examine chaotic behavior oflogistic map.

Quantum-like model of brain's functioning: Decision making from decoherence

We present a quantum-like model of decision making in games of the Prisoners Dilemma type. By this model the brain processes information by using representation of mental states in a complex Hilbert space. Driven by the master equation the mental state of a player, say Alice, approaches an equilibrium point in the space of density matrices (representing mental states). This equilibrium state determines Alices mixed (i.e., probabilistic) strategy. We use a master equation in which quantum physics describes the process of decoherence as the result of interaction with environment. Thus our model is a model of thinking through decoherence of the initially pure mental state. Decoherence is induced by the interaction with memory and the external mental environment. We study (numerically) the dynamics of quantum entropy of Alices mental state in the process of decision making. We also consider classical entropy corresponding to Alices choices. We introduce a measure of Alices diffidence as the difference between classical and quantum entropies of Alices mental state. We see that (at least in our model example) diffidence decreases (approaching zero) in the process of decision making. Finally, we discuss the problem of neuronal realization of quantum-like dynamics in the brain; especially roles played by lateral prefrontal cortex or/and orbitofrontal cortex.

Note on quantum probability

SummaryWhen 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.

Quantum ergodic channels in operator algebras

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

Quantum adaptivity in biology : From genetics to cognition

Preface Introduction Fundamentals of classical probability and quantum probability Theory Fundamentals of molecular biology Adaptive dynamics and general approach to non-Kolmogorov probability theory Application of adaptive dynamics to Biology Application to decision making theory and cognitive science Operational Approach to Modern Theory of Evolution Epigenetic Evolution and Theory of Open Quantum Systems Foundational Problems of Quantum Mechanics Decision and Intention Operators as Generalized Quantum Observables Index

NP Problem in Quantum Algorithm

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.

Entropy transmission in C*-dynamical systems

Abstract Three entropies of a state in C∗-dynamical systems are introduced and their relations and dynamical properties are studied. The entropy (information) transmission under a channel between two dynamical systems is considered. We find a condition under which our entropy becomes a dynamical invariant between two systems.

New quantum algorithm for studying NP-complete problems

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.