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

TL;DR

This paper constructs a gradient system on the quantum information space that models the Karmarkar flow, linking quantum information geometry with linear programming optimization methods.

Contribution

It introduces a novel gradient system on the quantum information space that realizes the Karmarkar flow, bridging quantum information geometry and linear programming.

Findings

Realizes the Karmarkar flow within the quantum information space.

Extends previous gradient systems to include the Karmarkar flow.

Provides a geometric framework connecting quantum information and optimization algorithms.

Abstract

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.