  In this work, we establish a so-called "system-bath entanglement theorem",
for arbitrary systems coupled with Gaussian environments. This theorem connects
the entangled system-bath response functions in the total composite space to
those of local systems, as long as the interacting bath spectral densities are
given. We validate the theorem with the direct evaluation via the exact
dissipaton-equation-of-motion approach. Therefore, this work enables various
quantum dissipation theories, which originally describe only the reduced system
dynamics, for their evaluations on the system-bath entanglement properties.
Numerical demonstrations are carried out on the Fano interference
spectroscopies of spin-boson systems.
