  We study a system coupled to external degrees of freedom, called bath, where
we assume that the total system, consisting of system and bath is in
equilibrium. An expansion in the coupling between system and bath leads to a
general form of the reduced density matrix of the system as a function of the
bath selfenergy. The coupling to the bath results in a renormalization of the
energies of the system and in a change of the eigenbasis. This theory is
applicable to quantum emulators in thermal equilibrium. Undesired external
degrees of freedom can affect their reliability. We study the influence of
bosonic degrees of freedom on the state of a six qubit system.
