Quantificating The Entanglement Of Quantum Graph State

Quantum information is the fusion of quantum mechanics, computer science, information theory and cryptography. Quantum entanglement is an indispensable physical resource in quantum information processing. Many application fields of quantum information theory such as quantum teleportation, quantum key distribution, quantum computing need to use this entanglement feature, so the study of entanglement is the core issue of quantum information theory. In practical applications, we not only need to know whether the given resource is entangled or not, but also need to confirm how much entanglement it contains, that is, the measure of entanglement.The quantification of entanglement has attracted wide attention in recent years, but the quantification of the entanglement receives a better solution only for bipartite quantum system, and the quantification of multipartite entanglement is still open even for a pure multipartite state. Until now, a variety of different entanglement measures have been proposed for multipartite setting, such as the Robustness of Entanglement, the Relative Entropy of Entanglement, and the Geometric Measure. However, all these methods involve variable complexity problem, which make the quantification of multipartite entanglement very difficult. Fortunately, it is hopeful to obtain the exact value of the multipartite entanglement of graph states, which are very usful multipartite quantum states in quantum information processing. Graph states are the specific algorithm resources for one-way quantum computing model, and they are subsets of stabilizer states which are widely used in quantum error correction. The three multipartie entanglement measures mentioned above are equal for a graph state.The entanglement quantification of graph state is relativly simple, for it can be described by graph language. So far, the study of graph state entanglement has just started, the latest research results is:determining the upper and lower bounds of graph state entanglement by using local operation and classical communication(LOCC), which can only confirm the entanglement of graph states that have equal bounds, but for graph states which have unequal bounds, it can only give a range of entanglement but not the exact value.The main contents of this paper are:1. We improve and simplify the methord of detemining the upper and lower bounds, and describe it by graph language.2. For the graph states which have unequal bounds, we propose a new method of entanglement calculation-iterative algorithm, and the precision of iteration algorithm of the entanglement is less than 10-14.3. We use the iterative algorithm to complete the entanglement quantification of 9 qubits, part of 10 qubits, and 11 qubits graph states, where 9 qubits graph states have 440 LC inequivalent graphs and 11 qubits graph states have fourty thousand. For graph states with diffent entanglement upper and lower bounds, the closest product states show different structural characteristics. We analyze the structural characteristics of the closest product states, and classify them as several types.