  Correlations lie at the heart of almost all scientific predictions. It is
therefore of interest to ask whether there exist general limitations to the
amount of correlations that can be created at a finite amount of invested
energy. Within quantum thermodynamics such limitations can be derived from
first principles. In particular, it can be shown that establishing correlations
between initially uncorrelated systems in a thermal background has an energetic
cost. This cost, which depends on the system dimension and the details of the
energy-level structure, can be bounded from below but whether these bounds are
achievable is an open question. Here, we put forward a framework for studying
the process of optimally correlating identical (thermal) quantum systems. The
framework is based on decompositions into subspaces that each support only
states with diagonal (classical) marginals. Using methods from stochastic
majorisation theory, we show that the creation of correlations at minimal
energy cost is possible for all pairs of three- and four-dimensional quantum
systems. For higher dimensions we provide sufficient conditions for the
existence of such optimally correlating operations, which we conjecture to
exist in all dimensions.
