Estimation of general Hamiltonian parameters via controlled energy measurements

The quantum Cramér-Rao theorem states that the quantum Fisher information bounds the best achievable precision in the estimation of a quantum parameter $\xi$. This is true, however, under the assumption that the measurement employed to extract information on $\xi$ is regular, i.e., neither its sample space nor its positive-operator valued elements depend on the (true) value of the parameter. A better performance may be achieved by relaxing this assumption. In the case of a general Hamiltonian parameter, i.e., when the parameter enters the system's Hamiltonian in a nonlinear way (making the energy eigenstates and eigenvalues $\xi$ dependent), a family of nonregular measurements, referred to as controlled energy measurements, is naturally available. We perform an analytic optimization of their performance, which enables us to compare the optimal controlled energy measurement with the optimal Braunstein-Caves measurement based on the symmetric logarithmic derivative. As the former may outperform the latter, the ultimate quantum bounds for general Hamiltonian parameters are different than those for phase (shift) parameters. We also discuss in detail a realistic implementation of controlled energy measurements based on the quantum phase estimation algorithm and work out a variety of examples to illustrate our results.

Figure 1

Comparison between an estimation strategy based on a controlled energy measurement (upper scheme) and one based on a regular measurement (lower scheme). In the first case, the opti- mal performance is quantified by ${\mathcal{G}}_{\xi }$, which is optimized over the preparation and control steps; in the second, it is quantified by the CQFI ${\mathcal{F}}_{\xi }^{(Q,C)}$, which instead is optimized over all initial preparations and (regular) measurements. The different stages of the schemes are denoted as follows: I. $\to$ preparation, II. $\to$ encoding, III. $\to$ control, IV. $\to$ energy measurement, and V. $\to$ regular measurement.

Figure 2

Circuit diagram of a simplified realistic controlled energy measurement with $n=4$ control qubits. A realistic controlled energy measurement replaces each application of ${\mathcal{C}}_{U_{\tau }}$ by $m$ repeated applications of ${\mathrm{\Gamma }}_{U_{\tau /m}}$, defined in Eq. (63).

Figure 3

Circuit diagram of $W_{U_{\tau /m}}$.

Figure 4

Comparison between the optimal Braunstein-Caves measurement and the optimal controlled energy measurement, for the estimation of the polar angular direction of a magnetic field via a qubit probe. The solid line is the CQFI, while the dashed line corresponds to ${\mathcal{G}}_{\xi }$, computed by Eq. (76). The circular marks denote ${\mathcal{G}}_{\xi }$, computed by numerical optimization, from its definition (14), thus confirming that the bound given in Proposition 1 is saturated.

Figure 5

Upper panel: FI of the best-performing realistic controlled energy measurement, for different values of $n$ and fixed $m=5$. Each marker represents the maximum FI, taken over the family of realistic controlled energy measurements for given $n$, $m$, $\tau$, and interrogation time $t$. The curves are obtained by interpolation. The thin solid curve corresponds to the QFI of Eq. (73). Notice that the optimal Braunstein-Caves measurement is outperformed already for $n=6$. Lower panel: FI of the best-performing realistic controlled energy measurement, for different values of $m$ and fixed $n=6$. Both plots are obtained for $\omega =1$ and $\tau =0.1$ (in the natural units of the problem), while the true value of the parameter is taken to be $\xi =\pi /4$.

Figure 6

Comparison between the optimal Braunstein-Caves measurement and the optimal controlled energy measurement, for the estimation of one component of a magnetic field via a qubit probe. The solid line is the CQFI, while the dashed line corresponds to ${\mathcal{G}}_{\xi }$. The circular marks denote ${\mathcal{G}}_{\xi }$, computed by numerical optimization, from its definition (14), thus confirming that the bound given in Proposition 1 is saturated.

Figure 7

Comparison between the optimal Braunstein-Caves measurement and the optimal controlled energy measurement for the estimation of the magnitude of a weak magnetic field via a NV center in diamond. The solid line is the CQFI, while the dashed line corresponds to ${\mathcal{G}}_{\xi }$. The circular marks denote ${\mathcal{G}}_{\xi }$, computed by numerical optimization, from its definition (14), thus confirming that the bound given in Proposition 1 is saturated.