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Journal of Applied Mathematics and Physics

Entanglement and Closest Product States of Graph States with 9 to 11 Qubits

Cuifeng Wang, Li-zhen Jiang, Lei Wang

Published Jan 1, 2013• 2 citations
Publisher

Abstract

The numbers of local complimentary inequivalent graph states for 9, 10 and 11 qubit systems are 440, 3132, 40457, respectively. We calculate the entanglement, the lower and upper bounds of the entanglement and obtain the closest product states for all these graph states. New patterns of closest product states are analyzed.

DOI
10.4236/jamp.2013.14010

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The numbers of local complimentary inequivalent graph
states for 9, 10 and 11 qubit systems are 440, 3132, 40457,
respectively. We calculate the entanglement, the lower and upper bounds of the
entanglement and obtain the closest product states for all these graph states.
New patterns of closest product states are analyzed.
The entanglement of graph states up to eight qubits is calculated in the regime of iteration calculation. The entanglement measures could be the relative entropy of entanglement, the logarithmic robustness or the geometric measure. All 146 local inequivalent graphs are classified as two categories: graphs with identical upper LOCC entanglement bound and lower bipartite entanglement bound, graphs with unequal bounds. The late may displays non-integer entanglement. The precision of iteration calculation of the entanglement is less than 10−1410^{-14}10−14.
The number of inequivalent classes of up to 8-qutrit graph states is 1002, with 239 decomposable graphs and 763 indecomposable graphs. Apparently, the former can be decomposed to several indecomposable parts. For qubit graph states, the upper and lower bounds of entanglement have been given. If the two bounds don't coincide, the entanglement can be calculated by iterative method. For qutrit graph states, their lower bounds can be found through the local complementation operations. But its hard to find the upper bound because the qutrit graph states have 2 basis states which are GHZ state and W state. Fortunately, the iterative method can also be used here. In this paper, we calculated the entanglement of all the 1002 graph states. Through researching the results, we found the similar feature with graph qubit states that the decimal part of entanglement is stable.
Graph states are multipartite entangled states that correspond to mathematical graphs, where the vertices of the graph now play the role of quantum multilevel systems and edges represent interactions of the systems. Graph states are the basis of quantum error correction and one-way quantum computer. We systematically study the entanglement of non-binary graph states. Using iterative algorithm and entanglement bounds, we calculate the entanglement of all the ternary graph states up to nine vertices and parts of quaternary and quinary graph states modulo local unitary transformations and graph isomorphisms. The entanglement measure can be the geometric measure, the measure of relative entropy of entanglement or the measure of logarithmic robustness. We classify the graph states according to the entanglement values obtained. The closest product states obtained in the calculations are studied.