A Class Of Linear Mappings Preserving Majorization

This paper does research on the linear mappings that preserves majorization at one fixing point in some spaces, basing on the relevant knowledge of linear algebra and quantum information, such as the explore on the doubly stochastic of corresponding matrix A, φ which is strictly isotone at one fixing point; the condition of the linear mapping that strictly isotone at one fixing point on R4is strictly isotone. There are four chapters.In Chapter1, it recalls the concepts about majorzation and local preserving linear mapping. In addition, it summarizes the developments of the recent study on it and the main contents of this paper. At last it introduces the research main content, purpose and meaning.In Chapter2, it mainly studies the linear mapping φ in fixed point (there is no equivalent component) strictly isotone, and preserving unit This matrix A which corresponded with this mapping has the doubly stochastic.In Chapter3, it mainly studies on the linear mapping fixed point of strictly isotone on R4.The first section applies the linear mapping preserving equivalent atei,(the ith element is1, the rest are0) corresponding to matrix A. Proves the matrix’s second, three or four column is the first column of the rearrangement.The second section it firstly applies a negative proposition that linear mapping preserving isotone at pointβ=(1,1,0,0)T preserves equivalence, which verifies the linear mapping that is strictly order preserving. And then it verifies that each linear mapping strictly in preserving unit and preserving equivalence at point e1is strictly order preserving at each fixed point β=(β1, β1,β2,β2). The third section applies a negative proposition that linear mapping preserving order at point β=(1,1,k,0)T (k≠1and k≠0) preserves equivalence, which verifies the linear mapping that is strictly order preserving. And then it verifies that each linear mapping strictly in preserving unit and preserving equivalence at point e1is strictly order preserving at each fixed pointβ=(β1,β1,β2,β3). At last, using the previous lemmas to prove that β is a strictly isotone at all point if and only if exist i≠j such thatβi≠βj,In Chapter four, it summarizes and gives some overviews in the future research.