The amount of correlation attainable between the components of a quantum system is constrained if the system is closed. We provide some examples, largely from the field of quantum thermodynamics, where knowing the maximal possible variation in correlations is useful. The optimization problem it raises requires us to search for the maximally and minimally correlated states on a unitary orbit, with and without energy conservation. This is fully solvable for the smallest system of two qubits. For larger systems, the problem is reduced to a manageable, classical optimization.
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