Quantum metrology paper in Quantum Science and Technology
Magda, Tillmann, and Animesh have had a paper accepted into Quantum Science and Technology (https://iopscience.iop.org/article/10.1088/2058-9565/aa7fa9) which looks at the trade-offs faced when attempting to simultaneously estimate phase and phase diffusion.
The quantum Cramér-Rao bound presents a lower bound on the attainable precision of parameter estimates from a quantum state, which is attainable for the estimation of any single parameter but not necessarily when estimating parameters simultaneously. In general a trade-off is introduced between attaining the best precision in one parameter while also needing to estimate others, which can be quantified by the total of the attainable error on each parameter over that predicted by the quantum Cramér-Rao bound.
Looking at states which have a fixed total number of particles they have shown that a fixed number state can attain the maximum possible trade-off in the limit of large phase diffusion. In the case of small phase diffusion they specifically considered Holland-Burnett states, which are derived from inputting a fixed number of photons into each input port of a balanced beam splitter and known to perform well for phase estimation, and found that this was a factor of two below the maximum trade-off possible in the limit that phase diffusion approaches zero.