构造纠缠目击的一般方法

General method of constructing entanglement witness

量子纠缠作为量子通信和量子计算过程中不可缺少的资源,在量子信息领域中有着广泛的应用.如何判定一个给定的量子态是否为纠缠态仍然是一个重要的课题.纠缠目击是一种特殊的自伴算子,它可以用来判断一个量子态是否为纠缠态.本文首先从纠缠目击的定义入手,给出构造纠缠目击的一般方法,证明了当一个可测量A在可分纯态上的最大期望CA严格小于它的最大特征值λ_(max)(A)时,对任何满足条件C_A≤C<λ_(max)(A)的参数C,算子W_C=CI-A都是一个纠缠目击;然后,作为应用得到了利用图态的稳定子构造纠缠目击的一系列方法.

Quantum entanglement, as an indispensable resource in quantum communication and quantum computation, is widely used in the field of quantum information. However, people＇s understanding on entanglement is quite limited both theoretically and experimentally. How to determine whether a given quantum state is entangled is still an important task. The entanglement witness is a kind of special self-adjoint operator, it can be used to determine whether a quantum state is an entangled state. This provides a new direction for the determination of entangled states. Entanglement witness has its own unique characteristics in various kinds of entanglement criterion. It is the most effective tool for detecting multipartite entanglement, and the most useful method to detect entanglement in experiments. In the background of quantum theory, we use theory of operators to make a thorough and systematic study of the construction of entanglement witness in this paper. First, from the definition of an entanglement witness, a general method is given to construct an entanglement witness. It is proved that when the maximal expectation CA of an observable A in the separable pure states is strictly less than its biggest eigenvalue λmax（A）, the operator We = CI-A is an entanglement witness provided that CA ≤ C 〈 λmax（A）. Although the entanglement witness WCA can detect more entangled states than Wc, but it is di^cult to calculate the exact value of CA, and the estimate of the upper bound of CA is easier. Therefore, it is more convenient to construct entanglement witness Wc than WCA. In quantum computation, a graph state is a special kind of multi-qubit state that can be represented by a graph. Each qubit is represented by a vertex of the graph, and there is an edge between every interacting pair of qubits. Graph states play a crucial role in many applications of quantum information theory, such as quantum error correcting codes, measurement-based quantum computation, and quantum simulation. Consequently, a significant effort is devoted to the creation and investigation of graph states. In the last part of this paper, as applications of our method, a series of methods for constructing an entanglement witness is obtained in the stabilizer formalism. It is also proved that how entanglement witnesses can be derived for a given graph state, provided some stabilizing operators of the graph state are known. Especially, when A is made up of some stabilizing operators of a graph state, entanglement witness WCA becomes one in literature.