Quantum technologies are developing powerful tools to generate and manipulate
coherent superpositions of different energy levels. Envisaging a new generation
of energy-efficient quantum devices, here we explore how coherence can be
manipulated without exchanging energy with the surrounding environment. We
start from the task of converting a coherent superposition of energy
eigenstates into another. We identify the optimal energy-preserving operations,
both in the deterministic and in the probabilistic scenario. We then design a
recursive protocol, wherein a branching sequence of energy-preserving filters
increases the probability of success while reaching maximum fidelity at each
iteration. Building on the recursive protocol, we construct efficient
approximations of the optimal fidelity-probability trade-off, by taking
coherent superpositions of the different branches generated by probabilistic
filtering. The benefits of this construction are illustrated in applications to
quantum metrology, quantum cloning, coherent state amplification, and
ancilla-driven computation. Finally, we extend our results to transitions where
the input state is generally mixed and we apply our findings to the task of
purifying quantum coherence.Comment: 35 pages, 10 figures; published versio
