Compatibility in multiparameter quantum metrology

Abstract

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.

Optimal Measurements for Simultaneous Quantum Estimation of Multiple Phases

“…However, unlike in the single-parameter case, the quantum Cramér-Rao bound is not always saturable, with a necessary condition provided by the commutativity of the symmetric logarithmic derivatives. Furthermore, it does not provide a recipe for constructing optimal measurements [3,31,34,36,37].…”

Monitoring dispersive samples with single photons: the role of frequency correlations

“…On the other hand, in the multi-parameter case, the quantum Cramér-Rao bound may not be achievable, as optimal measurements for di erent parameters may correspond to non-commuting observables. A necessary and su cient condition for simultaneous achievability of the quantum Cramér-Rao bound (3) is formulated in terms of the following weak-commutativity condition [8] Tr…”

“…While typically all the information needed can be e ciently collected through a single parameter [4,5], there are instances in which two parameters or more are necessary to capture the physical process under study [6][7][8]. Such parameters might not be associated to compatible observables, hence trade-o may appear in attempts at simultaneously measuring them at the ultimate quantum precision, especially when restrictions are imposed on the resources or, in other words, to the available Hilbert space [7,[9][10][11][12][13][14][15] These trade-o can be interpreted, and often circumvented, by understanding the estimation process under the geometrical standpoint by identifying the physical carrier of information with their state vectors [13]; however, quantum probes only partly approximate such geometric entities, since these typically describe one degree of freedom at the time.…”

Estimating phase with a random generator: Strategies and resources in multiparameter quantum metrology

“…However, in the best-case scenario, considering an estimation of, say, p parameters at once, the measurement precision of one parameter will be totally unaffected by the simultaneous measurement of the others, reducing the resources required by a factor of p. This is known as the parameters being compatible [17]. In general, exploring problems and developing solutions for multiparameter quantum metrology may not only result in an advantage in high-level applications such as microscopy, spectroscopy, optical or magnetic field sensing, or gravitational wave detection [7,18], but also provide deeper insights on multipartite quantum correlated states and quantum measurements.…”

“…However, one may also want to directly estimate the noise on the system, which may in turn allow for an improved estimation of the phase parameter, as well as being of interest in its own right. Rather than estimating the noise and phase parameters individually, some advantage may be attained by simultaneous estimation, bringing us into the field of multiparameter quantum metrology [15][16][17].…”