量子香农理论及其应用研究

Quantum Shannon theory is one of the most fundamental components of quantum information sciences. The main purpose of research is to combine information theory with quantum mechanics, and hence understand the ultimate performance limit of quantum information processing, such as transmission, storing, and extraction. Meanwhile, the methods and results developed in quantum Shannon theory, are applied widely to other research topics of quantum information sciences, serving as basic technical tools...Basing on our previous works, we suggest in this proposal an extensive study on the main problems of our concern in quantum Shannon theory. Our research includes the following two topics. The first one is the further development of quantum Shannon theory, with three goals: 1) understanding how the entropy changes during typical quantum processes; 2) investigating the capacity limit of quantum communication; 3) characterizing the decay rate of error in information extraction from quantum systems. The second one is the applications of quantum Shannon theory, to related topics such as quantum cryptography, quantum thermodynamics, and quantum computational complexity theory. Needless to say, quantum Shannon theory is a very active field of research, with major advances constantly being made, as well as significant problems remaining unsolved...We hope that, under the support of this fund, we will be able to promote greatly the investigation of quantum Shannon theory in China.

量子香农理论是量子信息科学的基础理论之一，其基本研究内容是将香农创立的信息论和量子力学相结合，理解量子信息的传输、存储、提取等处理过程的极限能力。同时，量子香农理论发展起来的方法和结果，被广泛应用于量子信息科学其它研究方向，是重要的理论工具。..本项目将在已有工作的基础上，就我们所关心的量子香农理论中的主要问题展开研究。我们的研究内容包括如下两个方面。一是量子香农理论的进一步发展，具体研究目标为：1）理解基本量子信息处理过程中熵的变化规律；2）探索量子通信的极限能力；3）刻画通过量子测量提取信息的误差收敛速率。二是量子香农理论在其它领域的应用，主要涉及量子密码学、量子热力学、量子计算复杂性理论等。毫无疑问，量子香农理论是一个充满活力的研究方向，不断地涌现出重大研究进展，而且有丰富的科学问题有待进一步解决。..我们期待，在本项目的支持下，大力推动量子香农理论在我国的发展。

量子香农理论有重要的理论和应用意义。本项目研究量子香农理论的几个核心内容。（1）可信函数：证明了量子极大相对熵光滑化、量子保密放大、量子信息解耦和相关的量子态合并任务的可信函数，给出了量子sandwiched Rényi 散度新的物理解释，即刻画量子信息处理任务收敛到最优情形的指数速率。（2）分数阶量子sandwiched Rényi散度的物理意义：通过证明量子极大相对熵光滑化、量子保密放大、量子信息解耦的强逆指数，首次发现了分数阶（阶数为1/2到1）的量子sandwiched Rényi 散度的物理意义。（3）量子信道的通信极限：获得了纠缠辅助下量子信道通信的强逆指数和量子信道模拟的直接错误指数，给出了量子信道的sandwiched Rényi互信息量的物理意义。

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