Multi-parameter metrology with the Holevo Cramer Rao bound; an explicit 2-qubit 3-parameter bound*

Quantum metrology is essential for the rigorous analysis and optimisation of all quantum-limited experiments. Since no experiment is ever truly a single parameter one what is required is multi-parameter quantum metrology. It invokes the central axiom of quantum mechanics, that of incompatible observables.

Hitherto, SLD quantum Fisher information matrix (QFIM) has largely been the Cramer-Rao bound of choice for examining multi-parameter metrology. In this work we show that the QFIM is an inadequate metric for multi-parameter problems. Further, we show that the Holevo Cramer-Rao (HCRB) bound is not only easier to calculate both numerically and analytically than previous thought but it is also essential for multi-parameter problems.

We point out new insights obtained from evaluating the HCRB for much larger Hilbert spaces than before, enabled by the convexity of the problem. We also show that previously known techniques allow us to calculate an explicit bound for 3D-magnetometry with 2 qubits, which again leads to new insight and further emphasises the pitfalls of the QFIM.