量子图态和图对角态的多组分纠缠

Multipartite entanglement is of fundamental importance in quantum information processing for various applications such as quantum computation, quantum error-correcting code and quantum secret sharing. It plays an important role in quantum phase transition, and is highly likely the reason for coherent energy transfer in photosynthetic light-harvesting complexes. Nowadays, experiments have succeeded in the preparation of 14-qubit multipartite entangled graph state. Due to noise and imperfect operations, usually the actual prepared state is a graph diagonal mixed state. Whether the prepared state is genuine multipartite entangled or not is of high interest. In contrast to the fast development of experiment results, theoretical research seems to be lagged far behind. It is only recently that a necessary and sufficient condition was proved for the genuine entanglement of a three-qubit Greenberger-Horne-Zeilinger (GHZ) graph diagonal state, and the similar condition for a four-qubit GHZ graph diagonal state was written down. The condition for fully separability of graph diagonal state is even more difficult. People can obtain part of results with numerical calculation for three-qubit GHZ graph diagonal state. Part of the theoretical challenge of understanding entanglement lies in the fact that it is also responsible for the quantum computational speedup: state space is big. The dimension of state space grows rapidly with the number of constituents in a composite quantum system. Graph state and graph diagonal state can be described with a smaller number of parameters, while still retaining the essential features of the multipartite entanglement problem. Any quantum state can be brought to graph diagonal state by a sequence of local operations the entanglement can not be increased in local operations, hence if the graph diagonal state is entangled, the original state should also be. Thus the research of multipartite entanglement of graph diagonal state is crucial both for experiment and theory. In this project, we will concentrate on the following critical topics: (1) a completely treatment on all the entanglement problems of three-qubit GHZ graph diagonal state, with emphasis on the analytical necessary and sufficient condition of full separability, Researches on the full separabilty of four-qubit GHZ and cluster diagonal states; (2) a systematically treatment on multipartite entanglement of graph states; (3) universality of expressions on the genuine multipartite entanglement of GHZ diagonal states, the necessary and sufficient conditions for genuine entanglement of graph diagonal states for seven or less qubit graphs;(4)the applications of graph diagonal states on the topics such as quantum capacity.

多组分纠缠是量子计算、量子纠错编码、量子秘密共享的基础,在量子相变中起重要作用,它也很有可能是光合作用中相干能量转移的关键. 目前实验进展迅速,已制备了多达14量子比特的多组分纠缠图态,因噪声和操作误差实际制备的是图对角的混合态,如何判断实验所得是否真多体纠缠乃所关心的问题,而理论研究滞后,现仅能判别3个和4个量子比特图对角态是否真纠缠,判别图对角态是否完全可分离就更困难.图态和图对角态具备多组分纠缠的复杂特征，同时可以用少量参数描述,且任意的多组分纠缠态通过局域操作可以转变为图对角态,因此图对角态纠缠的研究对实验和理论两方面都至关重要.本项目将(1)完全解决3组分图对角态的所有纠缠问题,研究4组分GHZ和簇对角态的完全可分离性,(2)系统研究图态多组分纠缠,(3)研究多组分GHZ对角态的真纠缠的普适性,部分7点以下图对角态的真纠缠充要条件,(4)研究图对角态在量子容量等方面的应用.

量子图态和图对角态是量子编码和量子容量的基础，同时在量子通信和量子计算中有广泛应用。.主要内容和结果：（一）、纠缠判据方面。(1)将可计算交叉模与重排准则用于连续变量系统的研究，得到高斯态纠缠准则，通过其高斯核的泛函导数得到非高斯态的纠缠准，以及一些非高斯态可分离充要条件。(2)基于量子特性函数，提出了一种新的纠缠判据，独立于其他纠缠准则，并具有很好的特性。(3)匹配纠缠见证者方法。给出了一个一般的方法，用以求出图对角态的k-可分离性凸集的准确边界。对于三体GHZ对角态的完全分离性得到充要条件，四体GHZ对角态的完全分离性和三分离性得到解析的纠缠条件，并且对交换对称态是充要的。对于一般的四体GHZ对角态和簇对角态，得到纠缠的充分条件。（二）、纠缠计算方面。(1) 论证了量子编码复杂度与量子码字纠缠的关系. 为研究量子码字的纠缠, 证明了三种纠缠测度对于量子稳定子码字是相等的, 纠缠的上下界可由量子编码的生成元确定. (2)研究了8量子比特及以下图态的真多组分纠缠的表征和量化问题. 对图态真纠缠, 证明三种纠缠测度是相等的. 给出了真多分纠缠的上界、下界和叠代算法. （三）、量子容量和通信应用方面. (1)只有量子简并码可以改进量子容量的散列界. 基于双色可着色图的图态提出一系列量子简并码. 对于消相位信道, 该类码的相干信息作为信道噪声的函数可以严格得到. 找到一种新的码, 其噪声容限高于重复码. (2) 应用量子图态级联编码,得到一般泡利信道在该编码输入下的多信道相干信息的公式,能够有效计算一般泡利信道量子容量的逼近值和信道传输量子信息的噪声容限。计算速度比Monte Carlo算法提高三个数量级. (3)设计了基于集体噪声信道三量子比特纠缠态测量相关性的鲁棒量子对话等协议。.本课题关于图态和图对角态纠缠的研究结果提示了一种渐进式添加的可重用的纠缠见证者库的思想方法，对多体系统的各种纠缠的准确检测有重要的意义。