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A successive approximation method for quantum separability
Deren HAN, Liqun QI
Front Math Chin. 2013, 8 (6): 1275-1293.
https://doi.org/10.1007/s11464-013-0274-1
Determining whether a quantum state is separable or inseparable (entangled) is a problem of fundamental importance in quantum science and has attracted much attention since its first recognition by Einstein, Podolsky and Rosen [Phys. Rev., 1935, 47: 777] and Schr?odinger [Naturwissenschaften, 1935, 23: 807-812, 823-828, 844-849]. In this paper, we propose a successive approximation method (SAM) for this problem, which approximates a given quantum state by a so-called separable state: if the given states is separable, this method finds its rank-one components and the associated weights; otherwise, this method finds the distance between the given state to the set of separable states, which gives information about the degree of entanglement in the system. The key task per iteration is to find a feasible descent direction, which is equivalent to finding the largest M-eigenvalue of a fourth-order tensor. We give a direct method for this problem when the dimension of the tensor is 2 and a heuristic cross-hill method for cases of high dimension. Some numerical results and experiences are presented.
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Neighbor sum distinguishing total colorings of K4-minor free graphs
Hualong LI, Bingqiang LIU, Guanghui WANG
Front Math Chin. 2013, 8 (6): 1351-1366.
https://doi.org/10.1007/s11464-013-0322-x
A total [k]-coloring of a graph G is a mapping ?: V (G) ∪E(G) → {1,2, ..., k} such that any two adjacent elements in V (G)∪E(G) receive different colors. Let f(v) denote the sum of the colors of a vertex v and the colors of all incident edges of v. A total [k]-neighbor sum distinguishing-coloring of G is a total [k]-coloring of G such that for each edge uv ∈E(G),f(u)≠f(v). By χnsd″(G), we denote the smallest value k in such a coloring of G. Pil?niak and Wo?niak conjectured χnsd″(G)≤?(G)+3 for any simple graph with maximum degree Δ(G). This conjecture has been proved for complete graphs, cycles, bipartite graphs, and subcubic graphs. In this paper, we prove that it also holds for K4-minor free graphs. Furthermore, we show that if G is a K4-minor free graph with ?(G)≥4, then χnsd″(G)≤?(G)+2 The bound Δ(G) + 2 is sharp.
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