Probably a trivial question. Given two random matrices $A, B$ such that $\left\langle \left[A,B\right]\right\rangle =0$, namely only the (element-wise) mean of the commutator is zero, can I say that $\left\langle e^{A+B}\right\rangle =\left\langle e^{A}e^{B}\right\rangle$? Eventually why?
Thanks in advance!