A gradient system on the quantum information space that realizes the Karmarkar flow for linear programming

In the paper of Uwano [Czech. J. of Phys., vol.56, pp.1311-1316 (2006)], a gradient system is found on the space of density matrices endowed with the quantum SLD Fisher metric (to be referred to as the quantum information space) that realizes a generalization of a gradient system on the space of multinomial distributions studied by Nakamura [Japan J. Indust. Appl. Math., vol.10, pp.179-189 (1993)]. On motived by those papers, the present paper aims to construct a gradient system on the quantum information space that realizes the Karmarkar flow, the continuous limit of the Karmarkar projective scaling algorithm for linear programming.