In this paper, sufficiently convenient conditions for the majorization order are given, shedding new light on the comparison of quantum systems. By using some technical lemmas, stochastic comparisons of density matrices under mixing operations and also stochastic comparisons of average magnitudes of nondecreasing energy functions are performed. Using newly defined order properties that imply the majorization property, the order property of quantum densities is determined using Kullback–Leibler relative entropy. Examples are given to illustrate the results.
