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