In quantum mechanics one postulates that for each state $i$ there is a matrix $A_i$ and for each measureable dynamical variable $j$ (velocity etc.) there is a matrix $B_j$. Both are Hermitian matrices over complex numbers.
The experimental average of the dynamical variable B is postulated to be
\text{'average of variable j with state i'}=\operatorname{Tr}(AB)
(additionally some restrictions are placed on the state density matrix $A\geq 0$ and $\operatorname{Tr}A=1$)
Does someone have an idea how this postulates restricts possible outcomes for the average values? Or can completely general systems from probability theory always be stated with this trace notation? Are there mathematically implicit correlations between different states or variables due to this postulate? Is there a similar framework in probability theory unrelated to QM?
All these questions are meant to be purely mathematically derived from the form of the equation.