The problem of estimating the frequency of a two-level atom in a noisy environment is studied. Our interest is to minimise both the energetic cost of the protocol and the statistical uncertainty of the estimate. In particular, we prepare a probe in a 'GHZ-diagonal' state by means of a sequence of qubit gates applied on an ensemble of n atoms in thermal equilibrium. Noise is introduced via a phenomenological time-non-local quantum master equation, which gives rise to a phase-covariant dissipative dynamics. After an interval of free evolution, the n -atom probe is globally measured at an interrogation time chosen to minimise the error bars of the final estimate. We model explicitly a measurement scheme which becomes optimal in a suitable parameter range, and are thus able to calculate the total energetic expenditure of the protocol. Interestingly, we observe that scaling up our multipartite entangled probes offers no precision enhancement when the total available energy is limited. This is at stark contrast with standard frequency estimation, where larger probes—more sensitive but also more 'expensive' to prepare—are always preferred. Replacing by the resource that places the most stringent limitation on each specific experimental setup, would thus help to formulate more realistic metrological prescriptions.
The problem of estimating the frequency of a two-level atom in a noisy environment is studied. Our interest is to minimise both the energetic cost of the protocol and the statistical uncertainty of the estimate. In particular, we prepare a probe in a 'GHZ-diagonal' state by means of a sequence of qubit gates applied on an ensemble of n atoms in thermal equilibrium. Noise is introduced via a phenomenological time-non-local quantum master equation, which gives rise to a phase-covariant dissipative dynamics. After an interval of free evolution, the n -atom probe is globally measured at an interrogation time chosen to minimise the error bars of the final estimate. We model explicitly a measurement scheme which becomes optimal in a suitable parameter range, and are thus able to calculate the total energetic expenditure of the protocol. Interestingly, we observe that scaling up our multipartite entangled probes offers no precision enhancement when the total available energy is limited. This is at stark contrast with standard frequency estimation, where larger probes—more sensitive but also more 'expensive' to prepare—are always preferred. Replacing by the resource that places the most stringent limitation on each specific experimental setup, would thus help to formulate more realistic metrological prescriptions.
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.
We consider frequency estimation in a noisy environment with noisy probes. This builds on previous studies, most of which assume that the initial probe state is pure, while the encoding process is noisy, or that the initial probe state is mixed, while the encoding process is noiseless. Our work is more representative of reality, where noise is unavoidable in both the initial state of the probe and the estimation process itself. We prepare the probe in a GHZ diagonal state, starting from n + 1 qubits in an arbitrary uncorrelated mixed state, and subject it to parameter encoding under dephasing noise. For this scheme, we derive a simple formula for the (quantum and classical) Fisher information, and show that quantum enhancements do not depend on the initial mixedness of the qubits. That is, we show that the so-called 'Zeno' scaling is attainable when the noise present in the encoding process is time inhomogeneous. This scaling does not depend on the mixedness of the initial probe state, and it is retained even for highly mixed states that can never be entangled. We then show that the sensitivity of the probe in our protocol is invariant under permutations of qubits, and monotonic in purity of the initial state of the probe. Finally, we discuss two limiting cases, where purity is either distributed evenly among the probes or concentrated in a single probe.
In recent years, several estimation strategies have been formulated to determine the value of an unknown parameter in the most precise way, taking into account the presence of noise. These strategies typically rely on the use of quantum entanglement between the sensing probes and they have been shown to be optimal in the asymptotic limit in the number of probes, as long as one performs measurements on shorter and shorter time scales. Here, we present a different approach to frequency estimation, which exploits quantum coherence in the state of each sensing particle in the long time limit and is obtained by properly engineering the environment. By means of a commonly used master equation, we show that our strategy can overcome the precision achievable with entanglement-based strategies for a finite number of probes. We discuss a possible implementation of the scheme in a realistic setup that uses trapped ions as quantum sensors.
A quantum system can be prepared in an entangled state that is very sensitive to changes introduced to its dynamics and thus potentially can be utilized as a precise sensor of dynamics parameters. N entangled two-level atoms evolving unitarily can be used for frequency estimation with Heisenberg-limited precision which scales as ${N}^{\ensuremath{-}2}$, whereas uncorrelated states give only a standard-limited precision which scales as ${N}^{\ensuremath{-}1}$. As entangled states are also sensitive to disturbances introduced by interactions with external environments, quantum enhancement of the precision may be significantly limited for open quantum systems. This is the case when an interaction with an environment leads to local Markovian dephasing and the precision scales as ${N}^{\ensuremath{-}1}$ for all atom preparations. For local non-Markovian dephasing, however, Matsuzaki et al. [Phys. Rev. A 84, 012103 (2011)] and Chin et al. [Phys. Rev. Lett. 109, 233601 (2012)] showed that the atoms prepared in the maximally entangled state achieve Zeno scaling $\ensuremath{\propto}\phantom{\rule{0.16em}{0ex}}{N}^{\ensuremath{-}3/2}$ in precision and argued that this limit should hold for any entangled state. Here we prove that Zeno scaling is indeed the best possible scaling for all local non-Markovian dephasing models which feature initial Zeno dynamics. Moreover, Zeno scaling can hold only for experiments in the Zeno-dynamics regime, otherwise scaling of the precision is standard.
Quantum entanglement and coherence often allow for protocols that outperform classical ones in estimating a system’s parameter. When using infinite-dimensional probes (such as a bosonic mode), one could, in principle, obtain infinite precision in a finite time for both classical and quantum protocols, which makes it hard to quantify potential quantum advantage. However, such a situation is unphysical, as it would require infinite resources, so one needs to impose some additional constraint: typically the average energy employed by the probe is finite. Here we treat both energy and time as a resource, showing that, in the presence of noise, there is a nontrivial interplay between the average energy and the time devoted to the estimation. Our results are valid for the most general metrological schemes (e.g., adaptive schemes, which may involve entanglement with external ancillae or any kind of continuous measurement). We apply recently derived precision bounds for all parameters characterizing the paradigmatic case of a bosonic mode, subject to Lindbladian noise. We show how the time employed in the estimation should be partitioned in order to achieve the best possible precision. In most cases, the optimal performance may be obtained without the necessity of adaptivity or entanglement with ancilla. We compare results with classical strategies. Interestingly, for temperature estimation, applying a fast-prepare-and-measure protocol with Fock states provides better scaling with the number of photons than any classical strategy.
We investigate strategies for reaching the ultimate limit on the precision of frequency estimation when the number of probes used in each run of the experiment is fixed. That limit is set by the quantum Cramér-Rao bound (QCRB), which predicts that the use of maximally entangled probes enhances the estimation precision, when compared with the use of independent probes. However, the bound is only achievable if the statistical model used in the estimation remains identifiable throughout the procedure. This in turn sets different limits on the maximal sensing time used in each run of the estimation procedure, when entangled and independent probes are used. When those constraints are taken into account, one can show that, when the total number of probes and the total duration of the estimation process are counted as fixed resources, the use of entangled probes is, in fact, disadvantageous when compared with the use of independent probes. In order to counteract the limitations imposed on the sensing time by the requirement of identifiability of the statistical model, we propose a time-adaptive strategy, in which the sensing time is adequately increased at each step of the estimation process, calculate an attainable error bound for the strategy, and discuss how to optimally choose its parameters in order to minimize that bound. We show that the proposed strategy leads to much better scaling of the estimation uncertainty with the total number of probes and the total sensing time than the traditional fixed-sensing-time strategy. We also show that, when the total number of probes and the total sensing time are counted as resources, independent probes and maximally entangled ones have now the same performance, in contrast to the nonadaptive strategy, where the use of independent is more advantageous than the use of maximally entangled ones. Published by the American Physical Society 2024