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)的表述和(2)基于量子系统动力熵的平均互熵表述。我们围绕上述(1)和(2)，得到了以下结果：(1)利用量子互熵型度量研究信息传输效率在通信过程中，通道具有将输入系统的状态传递到输出系统的功能，互熵表示输入系统通过通道正确地向输出系统发送的信息量。研究了量子互熵类型的度量，即Ohya互信息、相干信息和Lindblad-Nielsen熵，并对这些度量在有噪声的光信道中进行了比较。(2)量子系统动态熵的数学研究Connes-Stormer，Emch，Connes-Narnhofer-Thirring(CNT)，Alicki-Fannes(AF)，Ohya(Complex)，Accardi-Ohya-Watanabe(AOW)，Kossakowski-Ohya-Watanabe(KOW)等人研究了经典系统的动态熵(Kolmogorov-Sinai熵)推广到量子系统的尝试。在这项研究中，基于KOW动态熵的公式，我严格考察了量子动态熵的性能，计算了压缩输入态和噪声光信道以及由量子马尔可夫过程组成的动力学系统的广义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.