  There is current interest in investigating which variables play an important
role in the physical processes with an open composite quan- tum system that
ranges from the foundational issues to the tasks of diverse applications in
quantum physics and technology. In this paper we contrast the local versus the
collective classical-like behavior of a pair of interacting qubits immersed in
the mutually independent thermal baths. The qubits can be locally subjected to
any of the standard, microscopically modelled, generalized amplitude damping
and phase damping channel, as well as to the recently introduced generalized
depolarizing channel. As a criterion for the classical-like behavior we use the
least entropy production, i.e. the total correlation in the system. The
classical-like collective behavior is found for short time intervals for
strongly interacting qubits. Only in a certain trivial case we observe the
simultaneous collective and individual-qubits classical- ity. Observation
(non-observation) of the collective (individual) classicality can reveal the
kind and strength of the qubits interaction or the ratio of the qubits'
energy-gaps, while manipulating the qubits interaction (by tuning the strength
and/or refocussing) can give rise to the desired kind of classicality.
