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Only with the simultaneous estimation of multiple parameters are the quantumaspects of metrology fully revealed. This is due to the incompatibility ofobservables. The fundamental bound for multi-parameter quantum estimation isthe Holevo Cram\'er-Rao bound (HCRB) whose evaluation has so far remainedelusive. For finite-dimensional systems we recast its evaluation as asemidefinite program, with reduced size for rank-deficient states. We use thisresult to study phase and loss estimation in optical interferometry andthree-dimensional magnetometry with noisy multi-qubit systems. For the former,we show that, in some regimes, it is possible to attain the HCRB with theoptimal (single-copy) measurement for phase estimation. For the latter, we showa non-trivial interplay between the HCRB and incompatibility, and providenumerical evidence that projective single-copy measurements attain the HCRB inthe noiseless two-qubit case.

Evaluating the Holevo Cram\'er-Rao bound for multi-parameter quantum metrology