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Register a new userThe ultimate precision limits for noisy frequency estimation
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Quantum metrology protocols allow to surpass precision limits typical to classical statistics. However, in recent years, no-go theorems have been formulated, which state that typical forms of uncorrelated noise can constrain the quantum enhancement to a constant factor, and thus bound the error to the standard asymptotic scaling. In particular, that is the case of time-homogeneous (Lindbladian) dephasing and, more generally, all semigroup dynamics that include phase covariant terms, which commute with the system Hamiltonian. We show that the standard scaling can be surpassed when the dynamics is no longer ruled by a semigroup and becomes time-inhomogeneous. In this case, the ultimate precision is determined by the system short-time behaviour, which when exhibiting the natural Zeno regime leads to a non-standard asymptotic resolution. In particular, we demonstrate that the relevant noise feature dictating the precision is the violation of the semigroup property at short timescales, while non-Markovianity does not play any specific role.
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The ultimate precision in any measurement is dictated by the physical process implementing the observation. The methods of quantum metrology have now succeeded in establishing bounds on the achievable precision for phase measurements over noisy channels. In particular, they demonstrate how the Heisenberg scaling of the precision can not be attained in these conditions. Here we discuss how the ultimate bound in presence of loss has a physical motivation in the Kramers-Kronig relations and we show how they link the precision on the phase estimation to that on the loss parameter.
The ultimate precision limit in estimating the Larmor frequency of $N$ unentangled rotating spins is well established, and is highly important for magnetometers, gyroscopes and many other sensors. However this limit assumes perfect, single addressing, measurements of the spins. This requirement is not practical in NMR spectroscopy, as well as other physical systems, where a weakly interacting external probe is used as a measurement device. Here we show that in the framework of quantum nano-NMR spectroscopy, in which these limitations are inherent, the ultimate precision limit is still achievable using control and a finely tuned measurement.
In contrast to the standard quantum state tomography, the direct tomography seeks the direct access to the complex values of the wave function at particular positions (i.e., the expansion coefficient in a fixed basis). Originally put forward as a special case of weak measurement, it can be extended to arbitrary measurement setup. We generalize the idea of quantum metrology, where a real-valued phase is estimated, to the estimation of complex-valued phase, and apply it for the direct tomography of the wave function. It turns out that the reformulation can help us easily find the optimal measurements for efficient estimation. We further propose two different measurement schemes that eventually approach the Heisenberg limit. In the first scheme, the ensemble of measured system is duplicated and the replica ensemble is time-reversal transformed before the start of the measurement. In the other method, the pointers are prepared in special entangled states, either GHZ-like maximally entangled state or the symmetric Dicke state. In both methods, the real part of the parameter is estimated with a Ramsey-type interferometry while the imaginary part is estimated by amplitude measurements.
In an idealistic setting, quantum metrology protocols allow to sense physical parameters with mean squared error that scales as $1/N^2$ with the number of particles involved---substantially surpassing the $1/N$-scaling characteristic to classical statistics. A natural question arises, whether such an impressive enhancement persists when one takes into account the decoherence effects that are unavoidable in any real-life implementation. In this thesis, we resolve a major part of this issue by describing general techniques that allow to quantify the attainable precision in metrological schemes in the presence of uncorrelated noise. We show that the abstract geometrical structure of a quantum channel describing the noisy evolution of a single particle dictates then critical bounds on the ultimate quantum enhancement. Our results prove that an infinitesimal amount of noise is enough to restrict the precision to scale classically in the asymptotic $N$ limit, and thus constrain the maximal improvement to a constant factor. Although for low numbers of particles the decoherence may be ignored, for large $N$ the presence of noise heavily alters the form of both optimal states and measurements attaining the ultimate resolution. However, the established bounds are then typically achievable with use of techniques natural to current experiments. In this work, we thoroughly introduce the necessary concepts and mathematical tools lying behind metrological tasks, including both frequentist and Bayesian estimation theory frameworks. We provide examples of applications of the methods presented to typical qubit noise models, yet we also discuss in detail the phase estimation tasks in Mach-Zehnder interferometry both in the classical and quantum setting---with particular emphasis given to photonic losses while analysing the impact of decoherence.
We demonstrate that the sensitivity of high-precision pulsar timing experiments will be ultimately limited by the broadband intensity modulation that is intrinsic to the pulsars stochastic radio signal. That is, as the peak flux of the pulsar approaches that of the system equivalent flux density, neither greater antenna gain nor increased instrumental bandwidth will improve timing precision. These conclusions proceed from an analysis of the covariance matrix used to characterise residual pulse profile fluctuations following the template matching procedure for arrival time estimation. We perform such an analysis on 25 hours of high-precision timing observations of the closest and brightest millisecond pulsar, PSR J0437-4715. In these data, the standard deviation of the post-fit arrival time residuals is approximately four times greater than that predicted by considering the system equivalent flux density, mean pulsar flux and the effective width of the pulsed emission. We develop a technique based on principal component analysis to mitigate the effects of shape variations on arrival time estimation and demonstrate its validity using a number of illustrative simulations. When applied to our observations, the method reduces arrival time residual noise by approximately 20%. We conclude that, owing primarily to the intrinsic variability of the radio emission from PSR J0437-4715 at 20 cm, timing precision in this observing band better than 30 - 40 ns in one hour is highly unlikely, regardless of future improvements in antenna gain or instrumental bandwidth. We describe the intrinsic variability of the pulsar signal as stochastic wideband impulse modulated self-noise (SWIMS) and argue that SWIMS will likely limit the timing precision of every millisecond pulsar currently observed by Pulsar Timing Array projects as larger and more sensitive antennae are built in the coming decades.
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