Entanglement-enhanced estimation of a parameter embedded in multiple phases

Quantum-enhanced sensing promises to improve the performance of tasks using non-classical probes and measurements that require far fewer scene-modulated photons than best classical schemes, thereby granting previously-inaccessible information about a wide range physical systems. We propose generalized distributed framework uses an entangled quantum probe estimate scene-parameter encoded within array phases, with functional dependence on parameter determined by physics actual system. The receiver laser light source enhanced quantum-entangled multi-partite squeezed-vacuum phases desired scene-parameter. entanglement suppresses collective vacuum noise across phase array. report simple analytical expressions for Cram\'er Rao bound depend only optical model measured system, we show our structured asymptotically saturates Cram\'er-Rao in lossless case. Our approach enables Heisenberg limited precision estimating respect total energy, as well number modulated phases. Furthermore, study impact uniform loss system examine behavior both bounds. apply examples diverse radio-frequency phased-array directional radar, beam-displacement tracking atomic-force microscopy, fiber-based temperature gradiometry.