Coherence is associated with transient quantum states; in contrast, equilibrium thermal quantum systems have no coherence. We investigate the quantum control task of generating maximum coherence from an initial thermal state employing an external field. A completely controllable Hamiltonian is assumed allowing the generation of all possible unitary transformations. Optimizing the unitary control to achieve maximum coherence leads to a micro-canonical energy distribution on the diagonal energy representation. We demonstrate such a control scenario starting from a given Hamiltonian applying an external field, reaching the control target. Such an optimization task is found to be trap-less. By constraining the amount of energy invested by the control, maximum coherence leads to a canonical energy population distribution. When the optimization procedure constrains the final energy too tightly, local suboptimal traps are found. The global optimum is obtained when a small Lagrange multiplier is employed to constrain the final energy. Finally, we explore the task of generating coherences restricted to be close to the diagonal of the density matrix in the energy representation.
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