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.

An Inverse Matrix Formula in the RightQuantum Algebra The rightquantum algebra was introduced recently by Garoufalidis, Le and Zeilberger in their quantum generalization of the MacMahon master theorem. A bijective proof of this identity due to Konvalinka and Pak, and also the recent proof of the rightquantum Sylvesteru0027s determinant identity, make heavy use of a bijection related to the first fundamental transformation on words introduced by Foata. This paper makes explicit the connection between this transformation and rightquantum linear algebra identities; we give a new, bijective proof of the rightquantum matrix inverse theorem, we show that similar techniques prove the rightquantum Jacobi ratio theorem, and we use the matrix inverse formula to find a generalization of the (rightquantum) MacMahon master theorem.