Defining quantum dynamical entropy

Noboru Watanabe

In quantum information theory, Emch, Conne, and Stormer were the first who studied the complexity of quantum dynamical processes. After that, Ohya introduced the [Formula: see text]-mixing entropy for general quantum systems and he defined the mean entropy and the mean mutual entropy for quantum dynamical systems based on the [Formula: see text]-mixing entropy. Conne, Narnhoffer and Thirring introduced the dynamical entropy (CNT entropy) and several researchers discussed this concept. Alicki and Fannes defined a different dynamical entropy — AF entropy. In 1995, Voiculescu proposed the dynamical approximation entropy. Accardi, Ohya and Watanabe defined yet another dynamical entropy (AOW entropy) through a quantum Markov process in 1997. In 1999, Kossakowski, Ohya and Watanabe introduced the dynamical entropy (KOW entropy) with respect to completely positive maps. In this paper, we discuss the complexity of quantum dynamical processes to calculate the dynamical entropy for noisy optical channels. In order to discuss the efficiency of information communication processes, a measure of complexity of initial state itself and a measure of transmitted complexity through communication channels are necessary. Quantum entropies were formulated on the basis of the quantum probability theory. In quantum communication systems, von Neumann entropy and Ohya mutual entropy relate to these measures of complexities, respectively. Recently, several mutual entropy type measures (Lindblad-Nielsen entropy and coherent entropy) were defined making use of entropy exchange with respect to a channel and initial state. In this paper, we show which of the measures is the most suitable one for discussing the efficiency of information transmission for quantum processes.

Göran Lindblad

Dėnes Petz

Noboru Watanabe

In Ref. 27, Ohya proposed the information dynamics synthesizing several ways of studying complex systems. In information dynamics, there are two types of complexities, one is a complexity of state representing system itself and another is a transmitted complexity between two systems. Entropies in classical and quantum systems are examples of these complexities of information dynamics. Transmitted complexity is an important tool to analyze the efficiency of information transmission in communication processes. In order to treat a flow of dynamical process, dynamical entropies were introduced in not only classical but also quantum systems. Based on the transition expectation introduced by Accardi to study quantum Markov process, the KOW entropy for completely positive (CP) maps was defined in Ref. 17. The generalized AOW and the AF entropies was constructed by the KOW entropy. The compound states are important tool to define the transmitted entropy 43,44. The transmitted complexity associated with the separable compound states is defined by using the generalized AOW entropy in Refs. 36 and 45. In this paper, we define a transmitted complexity (quantum dynamical mutual entropy) by means of the modified compound states and we prove the fundamental inequalities of the transmitted complexity for the independent quantum dynamical systems.

Takayuki Miyadera, Masanori Ohya

Andrzej Kossakowski, Makoto Ohya, N. WATANABE

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.