Fundamental limits to frequency estimation: a comprehensive microscopic perspective

Abstract

We consider a metrology scenario in which qubit-like probes are used to sense an external field that affects their energy splitting in a linear fashion. Following the frequency estimation approach in which one optimizes the state and sensing time of the probes to maximize the sensitivity, we provide a systematic study of the attainable precision under the impact of noise originating from independent bosonic baths. Specifically, we invoke an explicit microscopic derivation of the probe dynamics using the spin-boson model with weak coupling of arbitrary geometry. We clarify how the secular approximation leads to a phase-covariant (PC) dynamics, where the noise terms commute with the field Hamiltonian, while the inclusion of non-secular contributions breaks the PC. Moreover, unless one restricts to a particular (i.e., Ohmic) spectral density of the bath modes, the noise terms may contain relevant information about the frequency to be estimated. Thus, by considering general evolutions of a single probe, we study regimes in which these two effects have a non-negligible impact on the achievable precision. We then consider baths of Ohmic spectral density yet fully accounting for the lack of PC, in order to characterize the ultimate attainable scaling of precision when N probes are used in parallel. Crucially, we show that beyond the semigroup (Lindbladian) regime the Zeno limit imposing the 1/N 3/2 scaling of the mean squared error, recently derived assuming PC, generalises to any dynamics of the probes, unless the latter are coupled to the baths in the direction perfectly transversal to the frequency encoding-when a novel scaling of 1/N 7/4 arises. As our microscopic approach covers all classes of dissipative dynamics, from semigroup to non-Markovian ones (each of them potentially non-phase-covariant), it provides an exhaustive picture, in which all the different asymptotic scalings of precision naturally emerge.

Noisy frequency estimation with noisy probes

“…Furthermore, in the article we have restricted ourselves to simple dephasing noise, it should be possible to generalise our findings for timeinhomogeneous, phase-covariant noise. It remains to be seen whether the n 1 5 6 scaling in [32] or n 1 7 8 scaling in [31] are also achievable with a noisy probe state.…”

“…Many of these studies are typically concerned with estimating the frequency ω rather than the phase f; the two parameters are related to each other by the encoding time t, i.e., f=ω t. Thus, the total running time of the encoding process itself is regarded as a resource [13]. These studies have shown that a super-extensive growth of the frequency sensitivity may still be attained under time-inhomogeneous, phase-covariant noise [26][27][28][29][30], and even more generic Ohmic dissipation [31], noise with a particular geometry [32,33], or setups related to quantum error correction [34][35][36]. See also [37,38] which question the role of entanglement in such schemes and give advice on practical implementations.…”

Quantum metrology in the presence of limited data

“…In particular, it can be seen as a distinguishability metric [20] that arises in the expansion of the Bures distance between two infinitesimally close states [3,18]. Moreover, its reciprocal gives us the asymptotic limit for the Bayesian mean square error as a function of the number of repetitions under some fairly general assumptions 3 [5] (see footnote 3), and this is also the case for other approaches that are more conservative than the Cramér-Rao bound too [21,22].Nevertheless, the fact that this technique normally requires many repetitions to be useful is an important drawback to study realistic physical systems such as those previously mentioned. This problem has already been acknowledged in the literature (e.g.…”

“…A conceptually simple and straightforward approach consists in using a general measure of uncertainty and estimating how many measurements are needed such that the results predicted by the asymptotic theory are valid, which can always be done numerically [5,28]. In addition, we can rely on numerical techniques such as Monte Carlo simulations [29] or machine learning [30] to perform the optimisation directly, or can simply examine the behaviour of the system when the number of resources is finite once we have established the asymptotic results [21]. This was precisely the idea behind the methodology proposed in [5], where we analysed the nonasymptotic performance of metrology protocols that had been optimised as if the asymptotic theory were valid, and we explored the structure of the non-asymptotic regime with concrete examples.A different possibility is to derive more general lower bounds that are valid in both the asymptotic and the non-asymptotic regimes, such as [31][32][33].…”

Energy-efficient quantum frequency estimation

“…Although the unavoidable effects of environmental noise often cancel out any quantum advantage [12][13][14][15][16], a super-extensive growth of the efficiency may still be attained under time-inhomogeneous phase-covariant noise [17][18][19][20], and even more generic Ohmic dissipation [21], noise with a particular geometry [22,23], or setups involving quantum error correction [24][25][26].…”