Compatibility in multiparameter quantum metrology

Simultaneous estimation of multiple parameters in quantum metrological models is complicated by factors relating to the (i) existence of a single probe state allowing for optimal sensitivity for all parameters of interest, (ii) existence of a single measurement optimally extracting information from the probe state on all the parameters, and (iii) statistical independence of the estimated parameters. We consider the situation when these concerns present no obstacle, and for every estimated parameter the variance obtained in the multiparameter scheme is equal to that of an optimal scheme for that parameter alone, assuming all other parameters are perfectly known. We call such models compatible. In establishing a rigorous theoretical framework for investigating compatibility, we clarify some ambiguities and inconsistencies present in the literature and discuss several examples to highlight interesting features of unitary and nonunitary parameter estimation, as well as deriving new bounds for physical problems of interest, such as the simultaneous estimation of phase and local dephasing.

Figure 1

(a) Simultaneous estimation of multiple parameters $({\phi }_1,\cdots ,{\phi }_p)$ based on results of a single measurement performed on the output of a quantum channel ${\mathrm{\Lambda }}_{{\phi }_1,\cdots ,{\phi }_p}$ acting on a single input probe $\rho$. (b) $p$ separate schemes where one estimates each parameter individually using dedicated probe states and measurements, treating in every run the remaining parameters as perfectly known. We say that the parameters $({\phi }_1,\cdots ,{\phi }_p)$ of the quantum channel estimation model are compatible if there exists a simultaneous estimation scheme where each parameter is estimated equally well as in a set of $p$ optimal schemes for the individual parameters, thus leading to a factor $p$ reduction in resources used.

Figure 2

Three scenarios of utilizing $n=N\nu$ quantum probes in metrology: (a) “classical” scheme, where both input probes and measurements are uncorrelated, resulting in $n$ independently and identically distributed random variables $x_i$; (b) entangled-probe scheme, where ${\rho }^N$ represents an arbitrary state on a Hilbert space ${\mathcal{H}}^{\otimes N}$, where $\mathcal{H}$ is the space upon which the channel ${\mathrm{\Lambda }}_{\mathbf{\phi }}$ acts, and measurements occur on the level of individual ${\rho }^N$; (c) collective measurement scheme, where input probes may be arbitrarily entangled and collective measurements over arbitrarily many ${\rho }^N$ are allowed.

Figure 3

A schematic of a general lossy interferometer with input state $\rho$. We model the losses by a beam splitter. In Ref. [22], a scheme was considered with transmissivity ${\eta }_2=1$, leading to one arm containing both the loss and phase parameters. We balance the interferometer by choosing ${\eta }_1={\eta }_2=\eta$.

Figure 4

Average $\xi$ of normalized uncertainties of estimating the phase and the dephasing parameter in the optimal simultaneous scheme (upper red line) and the optimal separate schemes (lower black line) as a function of the number of atoms used and the dephasing parameter set to $\eta =0.9$. For the separate schemes the considered average asymptotically saturates to 1, which is represented by black dashed line. The inset indicates the ratio of the precision achieved in both schemes indicating that the discrepancy is relatively small and decreases with increasing $N$ which indicates the possibility of satisfying the compatibility requirement in the asymptotic regime of large $N$.