Experimentally achieving the precision that standard quantum metrology schemes promise is always challenging. Recently, additional controls were applied to design feasible quantum metrology schemes. However, these approaches generally does not consider ease of implementation, raising technological barriers impeding its realization. In this paper, we circumvent this problem by applying closed-loop learning control to propose a practical controlled sequential scheme for quantum metrology. Purity loss of the probe state, which relates to quantum Fisher information, is measured efficiently as the fitness to guide the learning loop. We confirm its feasibility and certain superiorities over standard quantum metrology schemes by numerical analysis and proof-of-principle experiments in a nuclear magnetic resonance (NMR) system.
An entangled state prepared in the decoherence free sub-space together with a Ramsey type measurement can probe parity violation in heavy alkali ions like Ba+ or Ra+. Here we propose an experiment with Ba+ ions as an example to measure the small parity violating effect in this system.
Quantum metrology is a promising practical use case for quantum technologies, where physical quantities can be measured with unprecedented precision. In lieu of quantum error correction procedures, near term quantum devices are expected to be noisy, and we have to make do with noisy probe states. We prove that, for a set of carefully chosen symmetric probe states that lie within certain quantum error correction codes, quantum metrology exhibits an advantage over classical metrology even after the probe states are corrupted by a constant number of erasure and dephasing errors. These probe state…
We address open quantum systems out-of-equilibrium as effective quantum probes for the characterisation of their environment. We discuss estimation schemes for parameters driving a de-phasing evolution of the probe and then focus on qubits, establishing a relationship between the quantum Fisher information and the residual coherence of the probe. Finally, we apply our results to the characterisation of the ohmicity parameter of a bosonic environment.
Nonlinear interactions are recognized as potential resources for quantum metrology, facilitating parameter estimation precisions that scale as the exponential Heisenberg limit of $2^{-N}$. We explore such nonlinearity and propose an associated quantum measurement scenario based on the nonlinear interaction of $N$-probe entanglement generating form. This scenario provides an enhanced precision scaling of $D^{-N}/(N-1)!$ with $D > 2$ a tunable parameter. In addition, it can be readily implemented in a variety of experimental platforms and applied to measurements of a wide range of quantities, in…
An entangled state prepared in a decoherence free sub-space together with a Ramsey type measurement can probe parity violation in heavy alkali ions like Ba+ or Ra+. Here we propose an experiment with Ba+ as an example to measure the small parity violating effect in this system. It has been shown that a measurement on a maximally correlated system will reduce the uncertainty as compared to that on a single ion measurement. In addition it also provides a feasible solution to measure the nuclear spin dependent part of the total parity violating light shift in an ionic system which has so far not …
We introduce a positive Hermitian operator, the Fisher operator, and use it to examine a measurement process incorporating unitary dynamics and complete measurements. We develop the idea of information complement, the minimization of which establishes the optimal precision for a fixed input. The formalism demonstrates that, in general, the classical Fisher Information has the Hamiltonian semi-norm as an upper bound. This is achievable with a qubit probe and only projective measurements, and is independent of the true value of the estimated parameter. In an interferometry context, we show that …
In quantum metrology, quantum probe states are capable of estimating unknown physical parameters to precisions beyond classical limits. What qualities do such states possess? Here we relate the performance of a probe state at estimating a parameter $ϕ$ -- as quantified by the quantum Fisher information -- to the amount of purity it loses when $ϕ$ undergoes statistical fluctuation. This indicates that the better a state is for estimating $ϕ$, the more it decoheres when $ϕ$ is subject to noise.
A central feature of quantum metrology is the possibility of Heisenberg scaling, a quadratic improvement over the limits of classical statistics. This scaling, however, is notoriously fragile to noise. While for some noise types it can be restored through error correction, for other important types, such as dephasing, the Heisenberg scaling appears to be irremediably lost. Here we show that this limitation can sometimes be lifted if the experimenter has the ability to probe physical processes in a coherent superposition of alternative configurations. As a concrete example, we consider the prob…
It is well known that a quantum correlated probe can yield better precision in estimating an unknown parameter than classically possible. However, how such a quantum probe should be measured remains somewhat elusive. We examine the role of measurements in quantum metrology by considering two types of readout strategies: coherent, where all probes are measured simultaneously in an entangled basis; and adaptive, where probes are measured sequentially, with each measurement one way conditioned on the prior outcomes. Here we firstly show that for classically correlated probes the two readout strat…
We derive fundamental bounds on the maximal achievable precision in multiparameter noisy quantum metrology, valid under the most general entanglement-assisted adaptive strategy, which are tighter than the bounds obtained by a direct use of single-parameter results. This allows us to study the issue of the optimal probe incompatibility in the simultaneous estimation of multiple parameters in generic noisy channels, while so far the issue has been studied mostly in effectively noiseless scenarios (where the Heisenberg scaling is possible). We apply our results to the estimation of both unitary a…
The precision of quantum metrology is widely believed to be restricted by the Heisenberg limit, corresponding to a root mean square error that is inversely proportional to the number of independent processes probed in an experiment, N. In the past, some proposals have challenged this belief, for example using non-linear interactions among the probes. However, these proposals turned out to still obey the Heisenberg limit with respect to other relevant resources, such as the total energy of the probes. Here, we present a photonic implementation of a quantum metrology protocol surpassing the Heis…
The central issue in quantum parameter estimation is to find out the optimal measurement setup that leads to the ultimate lower bound of an estimation error. We address here a question of whether a Gaussian measurement scheme can achieve the ultimate bound for phase estimation in single-mode Gaussian metrology that exploits single-mode Gaussian probe states in a Gaussian environment. We identify three types of optimal Gaussian measurement setups yielding the maximal Fisher information depending on displacement, squeezing, and thermalization of the probe state. We show that the homodyne measure…
Quantum metrology theory has up to now focused on the resolution gains obtainable thanks to the entanglement among N probes. Typically, a quadratic gain in resolution is achievable, going from the 1/sqrt(N) of the central limit theorem to the 1/N of the Heisenberg bound. Here we focus instead on quantum squeezing and provide a unified framework for metrology with squeezing, showing that, similarly, one can generally attain a quadratic gain when comparing the resolution achievable by a squeezed probe to the best N-probe classical strategy achievable with the same energy. Namely, here we give a …
Quantum metrology pursues high-precision measurements of physical quantities by using quantum resources. However, the decoherence generally hinders its performance. Previous work found that the metrological error tends to diverge in the long-encoding-time regime due to the Born-Markovian approximate decoherence, which is called the no-go theorem of noisy quantum metrology. Here we propose a Gaussian quantum metrology scheme using bimodal quantized optical fields as the quantum probe. It achieves the precision of a sub-Heisenberg limit in the ideal case. However, the Markovian decoherence cause…
We derive a necessary and sufficient condition for the possibility of achieving the Heisenberg scaling in general adaptive multi-parameter estimation schemes in presence of Markovian noise. In situations where the Heisenberg scaling is achievable, we provide a semidefinite program to identify the optimal quantum error correcting (QEC) protocol that yields the best estimation precision. We overcome the technical challenges associated with potential incompatibility of the measurement optimally extracting information on different parameters by utilizing the Holevo Cramér-Rao (HCR) bound for pure …
We consider bipartite systems as versatile probes for the estimation of transformations acting locally on one of the subsystems. We investigate what resources are required for the probes to offer a guaranteed level of metrological performance, when the latter is averaged over specific sets of local transformations. We quantify such a performance via the average skew information, a convex quantity which we compute in closed form for bipartite states of arbitrary dimensions, and which is shown to be strongly dependent on the degree of local purity of the probes. Our analysis contrasts and comple…
We address the study of quantum metrology enhanced by indefinite causal order, demonstrating a quadratic advantage in the estimation of the product of two average displacements in a continuous variable system. We prove that no setup where the displacements are probed in a fixed order can have root-mean-square error vanishing faster than the Heisenberg limit 1/N, where N is the number of displacements contributing to the average. In stark contrast, we show that a setup that probes the displacements in a superposition of two alternative orders yields a root-mean-square error vanishing with super…
We derive families of optimal and near-optimal probe states for quantum estimation of the coupling constants of a general two-mode number-conserving bosonic Hamiltonian describing one-body and two-body dynamics. We find that the optimal states for estimating the dephasing of the modes, the self-interaction strength, and the contact interaction strength are related to the NOON states, whereas the optimal states for estimation of the intermode single particle tunneling amplitude are superpositions of antipodal SU(2) coherent states. For estimation of the amplitude of pair tunneling and the ampli…
Bounds on the ultimate precision attainable in the estimation of a parameter in Gaussian quantum metrology are obtained when the average number of bosonic probes is fixed. We identify the optimal input probe state among generic (mixed in general) Gaussian states with a fixed average number of probe photons for the estimation of a parameter contained in a generic multimode interferometric optical circuit, namely, a passive linear circuit preserving the total number of photons. The optimal Gaussian input state is essentially a single-mode squeezed vacuum, and the ultimate precision is achieved b…
Quantum metrology with entangled states in realistic noisy environments always suffers from decoherence. Therefore, the measurement precision is greatly reduced. Here we applied the dynamical decoupling method to protect the $N$-qubit quantum metrology protocol and successfully revived the scaling of the measurement precision as ${N^{-k}}$ with $k\in \left[{5/6,11/12}\right]$. The degree of the precision revival, as determined by the noise spectrum distribution, indicates that the performance of the protected protocol can be further improved by controlling the noise spectrum. Such a protected …
Quantum metrology overcomes standard precision limits and plays a central role in science and technology. Practically it is vulnerable to imperfections such as decoherence. Here, we demonstrate quantum metrology for noisy channels such that entanglement with ancillary qubits enhances the quantum Fisher information for phase estimation but not otherwise. Our photonic experiment covers a range of noise for various types of channels, including for two randomly alternating channels such that assisted entanglement fails for each noisy channel individually. We have simulated noisy channels by implem…
There is growing belief that the next decade will see the emergence of sensing devices based on the laws of quantum physics that outperform some of our current sensing devices. For example, in frequency estimation, using a probe prepared in an entangled state can, in principle, lead to a precision gain compared to a probe prepared in a separable state. Even in the presence of some forms of decoherence, it has been shown that the precision gain can increase with the number of probe particles $N$. Usually, however, the entangled and separable state preparation and readout times are assumed to be…
Quantum properties of the probes used to estimate a classical parameter can be used to attain accuracies that beat the standard quantum limit. When qubits are used to construct a quantum probe, it is known that initializing $n$ qubits in an entangled "cat state," rather than in a separable state, can improve the measurement uncertainty by a factor of $1/\sqrt{n}$. We investigate how the measurement uncertainty is affected when the individual qubits in a probe are subjected to decoherence. In the face of such decoherence, we regard the rate $R$ at which qubits can be generated and the total dur…
We show that for some noisy channels, the optimal entanglement-assisted strategy depends on the noise level. We note that there is a non-trivial crossover between the parallel-entangled strategy and the ancilla-assisted strategy - in the former the probes are all entangled, the latter the probes are entangled with a noiseless ancilla but not amongst themselves. The transition can be explained by the fact that, separable states are more robust against noise and therefore are optimal in the high noise limit, but they are in turn outperformed by ancilla-assisted ones.
Quantum metrology plays a fundamental role in many scientific areas. However, the complexity of engineering entangled probes and the external noise raise technological barriers for realizing the expected precision of the to-be-estimated parameter with given resources. Here, we address this problem by introducing adjustable controls into the encoding process and then utilizing a hybrid quantum-classical approach to automatically optimize the controls online. Our scheme does not require any complex or intractable off-line design, and it can inherently correct certain unitary errors during the le…
The problem of estimating an unknown phase $ φ$ using two-level probes in the presence of unital phase-covariant noise and using finite resources is investigated. We introduce a simple model in which the phase-imprinting operation on the probes is realized by a unitary transformation with a randomly sampled generator. We determine the optimal phase sensitivity in a sequential estimation protocol, and derive a general (tight-fitting) lower bound. The sensitivity grows quadratically with the number of applications $ N $ of the phase-imprinting operation, then attains a maximum at some $ N_\text{…
We investigate the role of coarsened measurement reference in quantum metrology. Coarsened measurement reference comes from the coarsened reference time and basis. When the measurement based on one common reference basis, the disadvantage can be removed by symmetry. Due to the coarsened reference basis, entangled states can not perform better than product states for large number of probe particles. Given a finite uncertainty of the coarsened reference basis, the optimal number of probe particle is obtained in estimating phase. Finally, we prove that the maximally entangled state always achieve…
Quantum-enhanced measurements exploit quantum mechanical effects to provide ultra-precise estimates of physical variables for use in advanced technologies, such as frequency calibration of atomic clocks, gravitational waves detection, and biosensing. Quantum metrology studies the fundamental limits in the estimation precision given a certain amount of resources (e.g. the number of probe systems) and restrictions (e.g. limited interaction time, or coping with unavoidable presence of noise). Here we show that, even in the presence of noise, probabilistic measurement strategies (which have a cert…
We use a small atomic Bose-Einstein condensate as an interferometric scanning probe to map out a microwave field near a chip surface with a few micrometers resolution. Using entanglement between the atoms we overcome the standard quantum limit of interferometry by 4 dB and maintain enhanced performance for interrogation times up to 20 ms. This demonstrates the usefulness of quantum metrology with entangled states when the particle number is limited due to the small probe size. Extending atom interferometry to micrometer spatial resolution enables new applications in electromagnetic field sensi…
We show why and when entanglement is needed for quantum-enhanced precision measurements, and which type of entanglement is useful. We give a simple, intuitive construction that shows how entanglement transforms parallel estimation strategies into sequential ones of same precision. We employ this argument to generalize conventional quantum metrology, to identify a class of noise whose effects can be easily managed, and to treat the case of indistinguishable probes (such as interferometry with light).
Quantum resources, such as entanglement, can decrease the uncertainty of a parameter-estimation procedure beyond what is classically possible. This phenomenon is well described for noiseless systems with asymptotically many measurement resources by the Quantum Cramer-Rao Bound, but no general description exists for the regime of limited measurement resources. We address this problem by defining a Bayesian quantifier for uncertainty suitable for the regime of limited resources, and by developing a mathematical description for a parameter-estimation procedure which uses qubit probes to estimate …
We address the relation between quantum metrological resolution and coherence. We examine this dependence in two manners: we develop a quantum Wiener-Kintchine theorem for a suitable model of quantum ruler, and we compute the Fisher information. The two methods have the virtue of including both the contributions of probe and measurement on an equal footing. We illustrate this approach with several examples of linear and nonlinear metrology. Finally, we optimize resolution regarding coherence as a finite resource.
Relativistic quantum metrology studies the maximal achievable precision for estimating a physical quantity when both quantum and relativistic effects are taken into account. We study the relativistic quantum metrology of temperature in (3+1)-dimensional de Sitter and anti-de Sitter space. Using Unruh-DeWitt detectors coupled to a massless scalar field as probes and treating them as open quantum systems, we compute the Fisher information for estimating temperature. We investigate the effect of acceleration in dS, and the effect of boundary condition in AdS. We find that the phenomenology of the…
The problem of quantum metrology under the context of a particular non-Markovian quantum evolution is explored. We study the dynamics of the quantum Fisher information (QFI) of a composite quantum probe coupled to a Lorentzian environment, for a full variety of different classes of parameters. We are able to find the best metrological state which is not maximally entangled but is the one which evolves fastest. This is shown by evidencing a connection between QFI and different quantum speed limits. At the same time, by optimizing a control field acting over the probes, we show how the total inf…
Scaling Enhancement in Quantum Metrology via Indefinite-Time-Direction Encoding
Shanghai Jiao Tong University（上海交通大学）；The University of Hong Kong（香港大学）；Hefei National Laboratory（合肥国家实验室）；Shanghai Research Center for Quantum Sciences（上海量子科学研究中心）
In this thesis we focus on Gaussian quantum metrology in the phase-space formalism and its applications in quantum sensing and the estimation of space-time parameters. We derive new formulae for the optimal estimation of multiple parameters encoded into Gaussian states. We discuss the discontinuous behavior of the figure of merit - the quantum Fisher information. Using derived expressions we devise a practical method of finding optimal probe states for the estimation of Gaussian channels and we illustrate this method on several examples. We show that the temperature of a probe state affects th…
Quantum metrology is an auspicious discipline of quantum information which is currently witnessing a surge of experimental breakthroughs and theoretical developments. The main goal of quantum metrology is to estimate unknown parameters as accurately as possible. By using quantum resources as probes, it is possible to attain a measurement precision that would be otherwise impossible using the best classical strategies. For example, with respect to the task of phase estimation, the maximum precision (the Heisenberg limit) is a quadratic gain in precision with respect to the best classical strate…
Quantum-enhanced metrology can be achieved by entangling a probe with an auxiliary system, passing the probe through an interferometer, and subsequently making measurements on both the probe and auxiliary system. Conceptually, this corresponds to performing metrology with the purification of a (mixed) probe state. We demonstrate via the quantum Fisher information how to design mixed states whose purifications are an excellent metrological resource. In particular, we give examples of mixed states with purifications that allow (near) Heisenberg-limited metrology, and provide example entangling H…
Multipartite quantum systems show properties which do not admit a classical explanation. In particular, even nonentangled states can enjoy a kind of quantum correlations called quantum discord. I discuss some recent results on the role of quantum discord in metrology. Given an interferometric phase estimation protocol where the Hamiltonian is initially unknown to the experimentalist, the quantum discord of the probe state quantifies the minimum precision of the estimation. This provides a physical interpretation to a widely investigated information-theoretic quantity.
The main goal of quantum metrology is to obtain accurate values of physical parameters using quantum probes. In this context, we show that abstention, i.e., the possibility of getting an inconclusive answer at readout, can drastically improve the measurement precision. We focus on phase estimation and quantify the required amount of abstention for a given precision. We also develop analytical tools to obtain the asymptotic behavior of the precision and required rate of abstention for arbitrary pure qubit states.
