  The main purpose of this paper is to establish a noncommutative analogue of
the Efron--Stein inequality, which bounds the variance of a general function of
some independent random variables. Moreover, we state an operator version
including random matrices, which extends a result of D. Paulin et al. [Ann.
Probab. 44 (2016), no. 5, 3431--3473]. Further, we state a Steele type
inequality in the framework of noncommutative probability spaces.
