Tight bounds on the simultaneous estimation of incompatible parameters

Quantum sensors are in the vanguard of new quantum technologies. The theoretical foundation of these sensors is quantum estimation theory, which provides fundamental bounds to the precision with which we can measure physical signals. A widely used limit is the quantum Cramér-Rao bound (QCRB).

It is well known that two signals may not be estimated simultaneously with their fundamental precision when the ideal estimation procedures for these signals are incompatible. The QCRB does not account for this incompatibility, and generates optimistic bounds that are unattainable. The Holevo Cramér-Rao bound (HCRB) provides a solution to this problem by optimising a single estimation procedure for all the signals. However, the HCRB has seen limited use in quantum estimation to date, since it involves a complex optimisation procedure, which was believed impossible to evaluate.

In this work, we analytically solve the HCRB for two-parameters and find an analytic expression for the optimal measurement procedure. This significantly advances the state of art of multi-parameter quantum estimation theory for incompatible observables. We apply our results to magnetometry and explore how bosonic quantum codes can bestow resilience of parameter estimates against noise beyond practical control.