Coherence as a Resource for Phase Estimation

Quantum phase estimation is a core task in quantum technologies ranging from metrology to quantum computing, where it appears as a key subroutine in various algorithms. Here, we quantitatively connect the performance of phase estimation protocols with quantum coherence. To achieve this, we construct and characterize resource theories of quantum networks that cannot generate coherence. Given multiple copies of a unitary encoding an unknown phase and access to a fixed coherent state, we estimate the phase using such networks. For a unified and general approach, we assess the quality of the estimate using a generic cost function that penalizes deviations from the true value. We determine the minimal average cost that can be achieved in this manner and explicitly derive optimal protocols. From this, we construct a family of coherence measures that directly connect a state's coherence with its value for phase estimation, demonstrating that every bit of coherence helps. This establishes coherence as a resource that quantifies the performance of phase estimation, and, thus, of any quantum technology relying on it as a subroutine.