Complementary quantum correlations among multipartite systems We study the monogamy and polygamy relations related to quantum correlations for multipartite quantum systems. General monogamy relations are presented for the (alpha )th ((0le alpha le gamma , gamma ge 2)) power of quantum correlation, and general polygamy relations are given for the (beta )th ((beta ge delta , 0le delta le 1)) power of quantum correlation. These monogamy and polygamy inequalities are complementary to the existing ones with different parameter regions of (alpha ) and (beta ). Applying these results to specific quantum correlations, the corresponding new classes of monogamy and polygamy relations are obtained, which include the existing ones as special cases. Detailed examples are given.

Selfconcordance and matrix monotonicity with applications to quantum entanglement problems Abstract   Let V be a Euclidean Jordan algebra and xcexa9 be a cone of invertible squares in V. Suppose that     g  :   R     xe2x86x92  R     is a matrix monotone function on the positive semiaxis which naturally induces a function      g  xcbx9c   :   xcexa9   xe2x86x92  V    . We show that     xe2x88x92   g  xcbx9c      is compatible (in the sense of Nesterovxe2x80x93Nemirovski) with the standard selfconcordant barrier     B  (  x  )    xe2x88x92  ln  det  (  x  )     on xcexa9. As a consequence, we show that for any cxc2xa0xe2x88x88xc2xa0xcexa9, the functions of the form     xe2x88x92  tr  (  c   g  xcbx9c    (  x  )   )    B   (  x  )      are selfconcordant on xcexa9. In particular, the function     x  xe2x86xa6  xe2x88x92  tr  (  c  ln  x  )     is a selfconcordant barrier function on xcexa9. Using these results, we apply a longstep pathfollowing algorithm developed in L.xc2xa0Faybusovich and C.xc2xa0Zhou Longstep pathfollowing algorithm for solving symmetric programming problems with nonlinear objective functions. Comput Optim Appl, 72(3):769795, 2019 to a number of important optimization problems arising in quantum information theory. Results of numerical experiments and comparisons with existing methods are presented.