Schur monotone increasing and decreasing sequences - Ganikhodzaev, Rasul; Saburov, Mansoor; Saburov, Khikmat (2013)
It is well known that on the one dimensional space, any bounded monotone increasing or monotone decreasing sequence converges to a unique limiting point. In order to generalize this result into the higher dimensional space, we should consider an appropriate order (or pre-order). A Majorization is a partial ordering on vectors which determines the degree of similarity between vectors. The majorization plays a fundamental role in nearly all branches of mathematics. In this paper, we introduce Schur monotone increasing and decreasing sequences on an n-dimensional space based on the majorization pre-order. We proved that the Cesaro mean (or an arithmetic mean) of any bounded Schur increasing or decreasing sequences converges to a unique limiting point. As an application of our result, we show that the Cesaro mean of mixing enhancing states of the quantum system becomes more stable and mixing than given states.