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arXiv:1712.08142 · New Journal of Physics

Noisy frequency estimation with noisy probes

Agnieszka Górecka, Felix A. Pollock, Pietro Liuzzo-Scorpo, Rosanna Nichols, Gerardo Adesso, Kavan Kishore Modi

Published Jul 20, 2018• 8 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1367-2630/aad4e5

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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.
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.
Dephasing noise is a ubiquitous source of decoherence in current atomic sensors. We address the problem of entanglement-assisted frequency estimation subject to classical dephasing noise with full spatial correlations (collective) and arbitrary temporal correlations. Our contributions are threefold. (i) We derive rigorous, state-independent bounds on the achievable estimation precision, showing how they are entirely determined by the short-time behavior of the decoherence function. For temporally uncorrelated (Markovian) dephasing, precision is limited by a probe-independent constant. For temporally correlated stationary noise, the bound approaches the noiseless limit for classical states, precluding any asymptotic quantum advantage. (ii) We show that these scaling bounds are tight, by constructing generalized Ramsey protocols that saturate them. These optimal protocols use squeezing at the input and before readout, both of which are available in state-of-the-art atomic interferometers. Implementing a perfect-echo protocol, which reaches Heisenberg scaling in the absence of noise, remains optimal in this noisy setting, irrespective of the noise temporal correlations. (iii) We prove that arbitrary collective open-loop control cannot lift the no-go for super-classical precision scaling under either Markovian or colored stationary noise, highlighting the detrimental nature of full spatial correlations. In the latter case, temporal correlations may nonetheless enable constant-factor improvements over the standard quantum limit, which may still be important in practical metrological scenarios.
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. NNN 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.
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.
Abstract We study the estimation precision attainable by entanglement-enhanced Ramsey interferometry in the presence of spatiotemporally correlated non-classical noise. Our analysis relies on an exact expression of the reduced density matrix of the qubit probes under general zero-mean Gaussian stationary dephasing, which is established through cumulant-expansion techniques and may be of independent interest in the context of non-Markovian open dynamics. By continuing and expanding our previous work (Beaudoin et al 2018 Phys. Rev. A 98 020102(R)), we analyze the effects of a non-collective coupling regime between the qubit probes and their environment, focusing on two limiting scenarios where the couplings may take only two or a continuum of possible values. In the paradigmatic case of spin–boson dephasing noise from a thermal environment, we find that it is in principle possible to suppress, on average, the effect of spatial correlations by randomizing the location of the probes, as long as enough configurations are sampled where noise correlations are negative. As a result, superclassical precision scaling is asymptotically restored for initial entangled states, including experimentally accessible one-axis spin-squeezed states.
We investigate different quantum parameter estimation scenarios in the presence of noise, and identify optimal probe states. For frequency estimation of local Hamiltonians with dephasing noise, we determine optimal probe states for up to 70 qubits, and determine their key properties. We find that the so-called one-axis twisted spin-squeezed states are only almost optimal, and that optimal states need not to be spin-squeezed. For different kinds of noise models, we investigate whether optimal states in the noiseless case remain superior to product states also in the presence of noise. For certain spatially and temporally correlated noise, we find that product states no longer allow one to reach the standard quantum limit in precision, while certain entangled states do. Our conclusions are based on numerical evidence using efficient numerical algorithms which we developed in order to treat permutational invariant systems.