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Phase measurement constitutes a key task in many fields of science, both in the classical and quantum regime. The higher precision of such measurement offers significant advances, and can also be utilised to achieve finer estimates for quantities such as distance, the gravitational constant, electromagnetic field amplitude, etc. Here we theoretically model the use of a quantum network, composed of a randomly coupled set of two-level systems, as a processing device for phase measurement. An incoming resource state carrying the phase information interacts with the quantum network, whose emission is trained to produce a desired output signal. We demonstrate phase precision scaling following the standard quantum limit, the Heisenberg limit, and beyond. This can be achieved using quantum resource states such as NOON states or other entangled states, however, we also find that classically correlated mixtures of states are alone sufficient, provided that they exhibit quantum coherence. Our proposed setup does not require conditional measurements, and is compatible with many different types of coupling between the quantum network and the phase encoding state, hence making it attractive to a wide range of possible physical implementations. Ministry of Education (MOE) Published version T.K., S.G., and T.C.H.L. acknowledge the support by the Singapore Ministry of Education under its AcRF Tier 2 Grant MOE2019-T2-1-004. T.P. is supported by the Polish National Agency for Academic Exchange NAWA Project No. PPN/PPO/2018/1/00007/U/00001. W.L. acknowledges support from the Foundation for Polish Science (IRAP project ICTQT, Contract No. 2018/MAB/5, co-financed by EU via Smart Growth Operational Programme). |
