A STUDY FOR ESTABLISHING THE FOUNDATION OF QUANTUM CORDING THEOREMS

为了讨论量子系统中的编码定理，我们需要研究(1)量子系统中互熵（信息）的表达式和(2)基于量子系统动态熵的平均互熵的表达式。对上述(1)和(2)进行聚焦，得到如下结果：(1)量子互熵型测量的信息传输效率研究在通信过程中，信道具有将输入系统的一种状态传递给输出系统的功能，而互熵表示从输入系统通过信道正确发送给输出系统的信息量。我研究了量子互熵类型的度量，即Ohya互熵、相干信息和Lindblad- Nielsen熵，并对这些度量进行了噪声光信道的比较。得到了Ohya互熵是最适合讨论量子编码定理的互熵。(2)量子系统动态熵的数学研究。cones - stormer、Emch、cones - narnhofer - thirring （CNT）、Alicki-Fannes （AF）、Ohya （Complexity）、accardii -Ohya- watanabe （AOW）、Kossakowski-Ohya-Watanabe（KOW）等人尝试将经典系统的动态熵（Kolmogorov-Sinai熵）扩展到量子系统。本研究在KOW动态熵公式的基础上，严格检验了量子动态熵的性能，计算了压缩输入态和噪声光通道以及由量子马尔可夫过程组成的动力系统的广义AOW熵。

In order to discuss coding theorems in quantum systems, we need to investigate (1) a formulation of mutual entropy (information) in quantum systems and (2) a formulation of mean mutual entropy based on dynamical entropy of quantum system. We focus the above (1) and (2), and we obtained the following results.(1)A study on efficiency of information transmission by means of quantum mutualentropy type measuresIn a communication process, a channel has a function to transmit a state of the input system to output system, and the mutual entropy expresses the amount of information correctly sent output system from input system through a channel. I studied the quantum mutual entropy type measures, that is Ohya mutual entropy, coherent information and Lindblad- Nielsen entropy and I compared with these measures for the noisy optical channel. I obtained the result that Ohya mutual entropy is most suitable one for discussing the quantum coding theorems.(2) A mathematical study of dynamical entropies for quantum systemsA trial of extension to quantum system of dynamic entropy (Kolmogorov-Sinai entropy) of classical system was studied by Connes-Stormer, Emch, Connes-Narnhofer-Thirring (CNT), Alicki-Fannes (AF), Ohya (Complexity), Accardi-Ohya-Watanabe (AOW), Kossakowski-Ohya-Watanabe(KOW), and so on. In this study, based on the formulation of KOW dynamical entropy, I strictly examined about the performance of the quantum dynamical entropy to calculate the generalized AOW entropy for the squeezed input state and the noisy optical channel, and for the dynamical system consisted of the quantum Markov process.