  Understanding system-bath correlations in open quantum systems is essential
for various quantum information and technology applications. Derivations of
most master equations (MEs) for the dynamics of open systems require
approximations that mask dependence of the system dynamics on correlations,
since the MEs focus on reduced system dynamics. Here we demonstrate that the
most common MEs indeed contain hidden information about explicit
system-environment correlation. We unfold these correlations by recasting the
MEs into a universal form in which the system-bath correlation operator
appears. The equations include the Lindblad, Redfield, second-order
time-convolutionless, second-order Nakajima-Zwanzig, and second-order universal
Lindblad-like cases. We further illustrate our results in an example, which
implies that the second-order universal Lindblad-like equation captures
correlation more accurately than other standard techniques.
