Quantum biology: Explore quantum dynamics in biological systems SCIENCE CHINA Information Sciences Quantum biology: explore quantum dynamics in biological systems SCIENCE CHINA Information Sciences 59, 081302 (2016); The modelling of quantum control systems Science Bulletin 60, 1493 (2015); Control problems in quantum systems

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Quantum information and artificial intelligence are the two most cutting-edge research fields in recent years, which have made a lot of progress in changing the traditional science, realizing the cross fusion of the two fields has become a hot topic of scientists. Although scholars have made many explorations in this field and used them to simulate the steady state and the dynamics of open quantum manybody systems, little attention has been paid to the problem of accurate representation of neural networks. In this paper, we focus on neural network representations of quantum mixed states. We first propose neural network quantum mixed virtual states (NNQMVS) and neural network quantum mixed states (NNQMS) with general input observables using two neural network architectures, respectively, and then explore their properties, the related conclusions of NNQMVS and NNQMS under tensor product operation and local unitary operation are obtained. To quantify the approximation degree normalized NNQMVS and NNQMS of a given mixed state, we define the best approximation degree using normalized NNQMVS and NNQMS, and obtain the necessary and sufficient conditions for the representability of a general mixed state using normalized NNQMVS and NNQMS. Moreover, we explore the types of mixed states that can be represented by these two neural network architectures and show their accurate neural network representations.

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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 max(A), the operator WC=CI-A is an entanglement witness provided that CA C max(A). Although the entanglement witness WCA can detect more entangled states than WC, but it is difficult 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.