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This is a tutorial aimed at illustrating some recent developments in quantumparameter estimation beyond the Cram\`er-Rao bound, as well as theirapplications in quantum metrology. Our starting point is the observation thatthere are situations in classical and quantum metrology where the unknownparameter of interest, besides determining the state of the probe, is alsoinfluencing the operation of the measuring devices, e.g. the range of possibleoutcomes. In those cases, non-regular statistical models may appear, for whichthe Cram\`er-Rao theorem does not hold. In turn, the achievable precision mayexceed the Cram\`er-Rao bound, opening new avenues for enhanced metrology. Wefocus on quantum estimation of Hamiltonian parameters and show that anachievable bound to precision (beyond the Cram\`er-Rao) may be obtained in aclosed form for the class of so-called controlled energy measurements. Examplesof applications of the new bound to various estimation problems in quantummetrology are worked out in some details.

Quantum enhanced metrology of Hamiltonian parameters beyond the Cram\`er-Rao bound