State change and entropies in quantum dynamical systems

Hiroki Majima, Michio Tokuyama, Irwin Oppenheim +1 more

The state of a system changes as a consequence of its internal dynamics and of the interaction with the environment that leads to a system‐environment correlation and cannot be represented in terms of unitary Hamiltonian dynamics of the system. In order to investigate the role of interaction between the system and its environment, we derive a general formula which relates the Hamiltonian of the system to entropy. This formula enables us to discriminate a type of interactions which causes the change in the state of the system in a spontaneous process. In other words, this formula enables us to know the type of interactions which would induce a new equilibrium state of the system in the process. It is found that the certain type of interactions does not affect the state (dynamics) of the system even if there are interactions between the system and its environment.

Noboru Watanabe

Noboru Watanabe, Andrei Yu. Khrennikov

Нобoрy Ватанабе, Noboru Watanabe

In 1989, Ohya propose a new concept, so-called Information Dynamics (ID), to investigate complex systems according to two kinds of view points.One is the dynamics of state change and another is measure of complexity.In ID, two complexities C S and T S are introduced.C S is a measure for complexity of system itself, and T S is a measure for dynamical change of states, which is called a transmitted complexity.An example of these complexities of ID is entropy for information transmission processes.The study of complexity is strongly related to the study of entropy theory for classical and quantum systems.The quantum entropy was introduced by von Neumann around 1932, which describes the amount of information of the quantum state itself.It was extended by Ohya for C*-systems before CNT entropy.The quantum relative entropy was first defined by Umegaki for σ-finite von Neumann algebras, which was 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 formulate 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 the entropic complexities for classical and quantum systems.We introduce some complexities by means of entropy functionals in order to treat the transmission processes consistently.We apply the general frames of quantum communication to the Gaussian communication processes.Finally, we discuss about a construction of compound states including quantum correlations.

Noboru Watanabe

Noboru Watanabe

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.