General Optimality of the Heisenberg Limit for Quantum Metrology

Quantum metrology promises improved sensitivity in parameter estimation over classical procedures. However, there is a debate over the question of how the sensitivity scales with the resources and the number of queries that are used in estimation procedures. Here, we reconcile the physical definition of the relevant resources used in parameter estimation with the information-theoretical scaling in terms of the query complexity of a quantum network. This leads to a completely general optimality proof of the Heisenberg limit for quantum metrology. We give an example of how our proof resolves paradoxes that suggest sensitivities beyond the Heisenberg limit, and we show that the Heisenberg limit is an information-theoretic interpretation of the Margolus-Levitin bound, rather than Heisenberg’s uncertainty relation.

Figure 1

(a) General parameter estimation procedure involving state preparation $P$, evolution $U(\phi )$, and generalized measurement $M$ with outcomes $x$, which produces a probability distribution $p(x|\phi )$. In terms of quantum networks, the evolution can be written as a number of queries of the parameter $\phi$. (b) Example for $N=4$ of the usual situation described by ${\mathcal{H}}_{\mathrm{GLM}}$ [the grey box represents $O_j(\phi )$]. (c) For ${\mathcal{H}}_{\mathrm{BFCG}}$ the number of queries $Q$ does not always equal the number of systems. (d) For ${\mathcal{H}}_{\mathrm{RB}}$ all possible subsets of systems perform a single query. The number of queries scales exponentially with the number of systems.