Abstract We study the impact of many-body effects on the fundamental precision limits in quantum metrology. On the one hand such effects may lead to nonlinear Hamiltonians, studied in the field of nonlinear quantum metrology, while on the other hand they may result in decoherence processes that cannot be described using single-body noise models. We provide a general reasoning that allows to predict the fundamental scaling of precision in such models as a function of the number of atoms present in the system. Moreover, we describe a computationally efficient approach that allows for a simple derivation of quantitative bounds. We illustrate these general considerations by a detailed analysis of fundamental precision bounds in a paradigmatic atomic interferometry experiment with standard linear Hamiltonian but with both single and two-body losses taken into account—a model which is motivated by the most recent Bose–Einstein condensate magnetometry experiments. Using this example we also highlight the impact of the atom number super-selection rule on the possibility of protecting interferometric protocols against decoherence.
Abstract We study the impact of many-body effects on the fundamental precision limits in quantum metrology. On the one hand such effects may lead to nonlinear Hamiltonians, studied in the field of nonlinear quantum metrology, while on the other hand they may result in decoherence processes that cannot be described using single-body noise models. We provide a general reasoning that allows to predict the fundamental scaling of precision in such models as a function of the number of atoms present in the system. Moreover, we describe a computationally efficient approach that allows for a simple derivation of quantitative bounds. We illustrate these general considerations by a detailed analysis of fundamental precision bounds in a paradigmatic atomic interferometry experiment with standard linear Hamiltonian but with both single and two-body losses taken into account—a model which is motivated by the most recent Bose–Einstein condensate magnetometry experiments. Using this example we also highlight the impact of the atom number super-selection rule on the possibility of protecting interferometric protocols against decoherence.
In an idealistic setting, quantum metrology protocols allow to sense physical parameters with mean squared error that scales as 1/N2 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 derive an explicit condition that determines whether in a noisy quantum frequency estimation problem the estimation precision of the most general adaptive quantum metrological protocol cannot reach the Heisenberg-like scaling. The condition is a simple algebraic statement on a relation between the Hamiltonian operator representing the unitary part of the dynamics and the noise operators appearing in the quantum Master equation, and does not require any finite-time integration of the dynamics. In particular these results allow us to understand when application of quantum error correction protocols in order to recover the Heisenberg scaling in quantum metrology is not possible. Additionally, we provide methods to obtain quantitative bounds on achievable precision in the most general adaptive quantum metrological models. Finally, we apply the newly developed tools to prove fundamental bounds in atomic interferometry with many-body effects such as many body losses as well as models involving many-body terms in the Hamiltonian part of the dynamics commonly referred to as non-linear quantum metrology.
We introduce general bounds for the parameter estimation error in nonlinear quantum metrology of many-body open systems in the Markovian limit. Given a k-body Hamiltonian and p-body Lindblad operators, the estimation error of a Hamiltonian parameter using a Greenberger-Horne-Zeilinger state as a probe is shown to scale as N^{-[k-(p/2)]}, surpassing the shot-noise limit for 2k>p+1. Metrology equivalence between initial product states and maximally entangled states is established for p≥1. We further show that one can estimate the system-environment coupling parameter with precision N^{-(p/2)}, while many-body decoherence enhances the precision to N^{-k} in the noise-amplitude estimation of a fluctuating k-body Hamiltonian. For the long-range Ising model, we show that the precision of this parameter beats the shot-noise limit when the range of interactions is below a threshold value.
Quantum precision enhancement is of fundamental importance for the development of advanced metrological optical experiments, such as gravitational wave detection and frequency calibration with atomic clocks. Precision in these experiments is strongly limited by the 1/√N shot noise factor with N being the number of probes (photons, atoms) employed in the experiment. Quantum theory provides tools to overcome the bound by using entangled probes. In an idealized scenario this gives rise to the Heisenberg scaling of precision 1/N. Here we show that when decoherence is taken into account, the maximal possible quantum enhancement in the asymptotic limit of infinite N amounts generically to a constant factor rather than quadratic improvement. We provide efficient and intuitive tools for deriving the bounds based on the geometry of quantum channels and semi-definite programming. We apply these tools to derive bounds for models of decoherence relevant for metrological applications including: depolarization, dephasing, spontaneous emission and photon loss. Quantum metrology employs the properties of quantum states to further enhance the accuracy of some of the most precise measurement schemes to date. Here, a method for estimating the upper bounds to achievable precision in quantum-enhanced metrology protocols in the presence of decoherence is presented.
Quantum metrology offers enhanced performance in experiments on topics such as gravitational wave-detection, magnetometry or atomic clock frequency calibration. The enhancement, however, requires a delicate tuning of relevant quantum features, such as entanglement or squeezing. For any practical application, the inevitable impact of decoherence needs to be taken into account in order to correctly quantify the ultimate attainable gain in precision. We compare the applicability and the effectiveness of various methods of calculating the ultimate precision bounds resulting from the presence of decoherence. This allows us to place a number of seemingly unrelated concepts into a common framework and arrive at an explicit hierarchy of quantum metrological methods in terms of the tightness of the bounds they provide. In particular, we show a way to extend the techniques originally proposed in Demkowicz-Dobrzański et al (2012 Nature Commun. 3 1063), so that they can be efficiently applied not only in the asymptotic but also in the finite number of particles regime. As a result, we obtain a simple and direct method, yielding bounds that interpolate between the quantum enhanced scaling characteristic for a small number of particles and the asymptotic regime, where quantum enhancement amounts to a constant factor improvement. Methods are applied to numerous models, including noisy phase and frequency estimation, as well as the estimation of the decoherence strength itself.
Quantum metrology fundamentally relies upon the efficient management of quantum uncertainties. We show that under equilibrium conditions the management of quantum noise becomes extremely flexible around the quantum critical point of a quantum many-body system: this is due to the critical divergence of quantum fluctuations of the order parameter, which, via Heisenberg's inequalities, may lead to the critical suppression of the fluctuations in conjugate observables. Taking the quantum Ising model as the paradigmatic incarnation of quantum phase transitions, we show that it exhibits quantum critical squeezing of one spin component, providing a scaling for the precision of interferometric parameter estimation which, in dimensions d>2, lies in between the standard quantum limit and the Heisenberg limit. Quantum critical squeezing saturates the maximum metrological gain allowed by the quantum Fisher information in d=∞ (or with infinite-range interactions) at all temperatures, and it approaches closely the bound in a broad range of temperatures in d=2 and 3. This demonstrates the immediate metrological potential of equilibrium many-body states close to quantum criticality, which are accessible, e.g., to atomic quantum simulators via elementary adiabatic protocols.