Bipartite systems

At the current time, there is no simple way to carry out the calculations with all these entanglement measures. Their properties, such as additivity, convexity, and continuity, and relationships are still under active investigation. Even for the best-understood entanglement of formation of the mixed states in bipartite systems AB, once the dimension or A or B is three or above, we don t know how to express it simply, although we have the general definitions given previously. However, for the case where both subsystems A and B are spin-i particles, there exists a simple formula from which the entanglement of formation can be calculated [42]. 

For the sake of simplicity, we limit ourselves to extended half-filled bipartite systems, with equal number of sites in each sublattice, A (non-starred) and B (starred),... 

Along with the primitive translations, and glide-reflections when appropriate, there axe other symmetry operations belonging to the space group. In bipartite systems, it is relevant to classify any symmetry operation according to whether it leaves each sublattice invariant or transforms one into each other (see, for instance, Fig. 2). 

In a bipartite system a set of d N independent singlets can be constructed by pairing to a singlet each of the N spins in A to a spin in B, with no site unpaired. We represent one of these spin-pairings (SP) by an arrow from the site in the sublattice A to its partner in B (see, for instance, Fig. 3, where a complete set of linearly-independent singlets for six-site systems are represented). 

For translationally symmetric bipartite systems with NA = NB = N and cyclic boundary conditions, either bn is zero or bn = — 6n-i, although actual values of bn for a given strip depend to a certain extend on the unit cell selected. For instance, bn is even for ladders with an even number of legs, while bn is odd for ladders with an odd number of legs. In particular,... 

Indeed for the Neel-based theory to work best it is better to have a bipartite system (i.e., a system with two sets of sites all of either set having solely only members from the other set as neighbors). Of course, when there is a question about the adequacy of the zero-order description questions about the (practical) convergence of the perturbation series arises. But for favorable systems these [38] or closely related [41] expansions can now be made through high orders to obtain very accurate results. 

As we mentioned before, when a biparticle quantum system AB is in a pure state, there is essentially a unique measure of the entanglement between the subsystems A and B given by the von Neumann entropy S = —Tr[p log2 PaI- This approach gives exactly the same formula as the one given in Eq. (26). This is not surprising since all entanglement measures should coincide on pure bipartite states and be equal to the von Neumann entropy of the reduced density matrix (uniqueness theorem). 

Since a benzenoid system H is a bipartite graph, the existence of Kekule structures of H is equivalent to the existence of 1-factors (perfect matchings) of a bipartite graph. In 1935, P. Hall found the following necessary and sufficient conditions. 

Clearly, if X = Y a matching which saturates every vertex in X is a 1-factor (perfect matching) of G. Hence Theorem 1 can be used to decide whether or not a given benzenoid system H has Kekule structures. But, using Theorem 1, we have to examine all subsets of X, where (X, Y)is the bipartition of H. This is evidently tedious. 

If all the elementary reactions are monomolecular, i.e. can be written as Ax —> Aj, it is more convenient to represent reaction mechanisms in a different way, namely nodes correspond to substances, edges are elementary reactions, and edge directions are the directions of reaction processes. As usual, this graph is simpler than the bipartite graph. For example, for the system of three isomers Al A2 and A3 we obtain... 

Consider the Al-S system shown in Figure 11.8. This phase diagram displays several features that are typical of many binary metal chalcogenides. There is only one stoichiometric line compound (bipartite phase) at room temperature, AI2S3. However, at one atm. 

Topological matrix of a bipartite graph. Bipartite graphs, just as the corresponding alternant systems, possess a number of remarkable properties. In particular, their vertices can always be enumerated so that the topological matrix is simplified and reduced to the block form... 

QUBIT ENTANGLEMENT FROM A BIPARTITE ATOMIC SYSTEM UNDER STRONG ATOM-VACUUM-FIELD COUPLING IN A CARBON NANOTUBE... 

ABSTRACT This paper explores the strengths and effectiveness of the Norwegian industrial relations system with respect to safety. How does industrial relations system in general and tripartism/bipartism in particular affect safety performance within oil and gas and metal industry By drawing upon qualitative data from the Norwegian offshore based oil- and gas industry on the one hand, and the onshore metal industry on the other, we discnss how the practice of tripartite collaboration imfolds in these two industries with respect to collaboration on increasing the safety level and the safety awareness on company level. The conclusion is that in the oil-and gas sector tripartite collaboration is institutionahzed into arenas where safety is scrutinized, while in the metal industry there is no such tripartite collaboration, only bipartite union-management collaboration on local plant level. 

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