量子态制备与识别的数学基础研究

In quantum communication, information is encoded by quantum states. The writing and reading process from encoded quantum states corresponds to preparation and discrimination of quantum states. According to quantum no-cloning theorem, the principle of state superposition, and quantum demolition measurement, it is impossible to prepare a state perfectly and distinguish non-orthogonal states deterministically. As a result, the core problem of quantum information processing is to explore better methods to prepare and distinguish quantum states. In this project, we will study the core problem on the background of quantum correlation, which are mostly researched recently, by utilizing automata theory, matrix theory, convex optimization, graph theory and so on,. The project consists of the following parts: 1. Based on matrix decomposition theory and quantum automata theory, we will study quantum state local preparation and the corresponding quantum circuits. 2. In quantum communication with general quantum correlation, quantum state preparation between long distance parties will be studied. The applied preparation methods include teleportation, telecloning and remote information concentration, remote state preparation. 3. Study on quantum state discrimination, including, minimum-error discrimination and unambiguous discrimination of states in isolated system, quantum state discrimination using local operation and classical communication, and quantum state discrimination by many observers in a sequential way..Based on the study, we want to explore the analytical relation of fidelity, successful probability, quantum correlation and measurement operators in the process of state preparation and discrimination. So that the general framework of quantum state preparation and discrimination can be constructed, The result of the study may have potential applications in future quantum communication.

在量子通信中，信息的载体是量子态，其写入和读取过程对应于量子态的制备和识别。但受到量子态不可克隆原理，态叠加原理，以及测量会破坏量子态的限制，量子态的制备和非正交态的识别不能完美进行。因此，探寻更好的量子态制备和识别方法一直是量子信息处理需要解决的核心问题。本项目将以量子关联为背景，利用量子自动机理论、矩阵理论、图论、凸优化等工具，研究量子态的制备和识别，主要包括：1. 基于矩阵分解理论和量子自动机最小化的量子态当地制备及其线路研究；2. 量子关联作为共享资源的量子态远距离制备，制备方式包括隐形传态、远程克隆和集中、远程态制备等；3. 量子态识别研究，包括独立系统一般态的最小错误和无歧义识别、利用局域操作和经典通信的识别、多个观测者按顺序识别等。.通过这些研究，探讨制备保真度、识别成功率和量子关联、测量算子等之间的解析关系，建立量子态制备与识别的框架结构理论，应用于量子通信。

由于量子信息依附的实体是量子系统，表示为复希尔伯特空间的单位向量，因此量子信息处理过程实际上要比经典信息处理复杂得多。量子态的制备和传输是量子信息处理过程中的两个基本环节，但是受量子力学不可克隆原理的限制，要想准确无误的拷贝信息，识别两个非正交的量子态都是不可能的。项目组在国家自然科学基金面上项目的资助下，从数学理论角度入手，逻辑推理提出了量子态制备和传输的各种方法和通信模型，有助于提高量子网络通信过程的信息存储准确率和传输成功率。主要研究成果如下:.1.提出了一种未知任意二粒子的双向受控量子隐形传态的方法，并获得发明专利授权。该发明能实现量子态的双向传输，且具有较高的量子信息传输容量和传输效率。.2.针对三方量子系统，设计了一种三方联合远程态制备方案，严格证明了方案的逻辑可行性，达到三个单量子态在不同地点同时制备的目的。方案降低了测量操作难度，减小了幅值阻尼噪声的影响。.3.利用χ型纠缠态的非局域关联特性，设计了一种安全的同时密集编码方案。方案中以双受控非门(DCNOT)算子锁定信道，从而达到两个接收者同时解码经典信息的效果，提高了协议的安全性。.4.有关成果可用于涉及通信安全的金融、电信、电力、电子信息等领域。相关研究取得发明专利授权2项，发表论文36篇。这些研究进一步拓展和丰富了量子信息相关理论和技术基础。