Abstract: Cauchy′s inequality has an important application in the study of unitary uncertainty. The authors use Cauchy′s inequality to improve itself and apply it to the calculation of unitary uncertainty. Firstly，the idea of using low-dimensional Cauchy′s inequality to optimize high-dimensional Cauchy′s inequality is used to obtain the first sequence of Cauchy′s inequality. Then， by introducing a convex function， the second sequence of Cauchy′s inequality is obtained. Finally，the obtained sequence of Cauchy′s inequality is applied to study the variance-based quantum uncertainty. The results show that the Cauchy′s inequality sequence constructed by us can effectively optimize the bounds of variance product-based uncertainty for two unitary operators.

Key words: Cauchy′s inequality, Cauchy′s function, convex function, unitary uncertainty