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Phase diffusion invariably accompanies all phase estimation strategies --quantum or classical. A precise estimation of the former can often providevaluable understanding of the physics of the phase generating phenomena itself.We theoretically examine the performance of fixed-particle number probe statesin the simultaneous estimation of phase and collective phase diffusion. Wederive analytical quantum limits associated with the simultaneous localestimation of phase and phase diffusion within the quantum Cramer-Rao boundframework in the regimes of large and small phase diffusive noise. The formeris for a general fixed-particle number state and the latter for Holland Burnettstates, for which we show quantum-enhanced estimation of phase as well as phasediffusion. We next investigate the simultaneous attainability of these quantumlimits using projective measurements acting on a single copy of the state interms of a trade-off relation. In particular, we are interested how thistrade-off varies as a function of the dimension of the state. We derive ananalytical bound for this trade-off in the large phase diffusion regime for aparticular form of the measurement, and show that the maximum of 2, set by thequantum Cramer-Rao bound, is attainable. Further, we show numerical evidencethat as diffusion approaches zero, the optimal trade-off relation approaches 1for Holland-Burnett states. These numerical results are valid in the smallparticle number regime and suggest that the trade-off for estimating oneparameter with quantum-limited precision leads to a complete lack of precisionfor the other parameter as the diffusion strength approaches zero. Finally, weprovide numerical results showing behaviour of the trade-off for a generalvalue of phase diffusion when using Holland-Burnett probe states.

Reaching for the quantum limits in the simultaneous estimation of phase and phase diffusion