Characterizing a system often demands learning its response function to an applied field. Such knowledge is rooted on the experimental evaluation of punctual fiducial response and interpolation to access prediction at arbitrary values. Quantum metrological resources are known to provide enhancement in assessing these fiducial points, but the implications for improved function estimation have only recently been explored, and have not been yet demonstrated. Here we show an experimental realization of function estimation based on a photonic achitecture. The phase response of a liquid-crystal to a voltage has been reconstructed by means of quantum and classical phase estimation, providing evidence of the superiority of the former and highlighting the interplay between punctual statistical error and interpolation error. Our results show how quantum resources should successfully be employed to access the rich information contained in continuous signals.
We present a framework for the quantum enhanced estimation of multiple parameters corresponding to non-commuting unitary generators. Our formalism provides a recipe for the simultaneous estimation of all three components of a magnetic field. We propose a probe state that surpasses the precision of estimating the three components individually and discuss measurements that come close to attaining the quantum limit. Our study also reveals that too much quantum entanglement may be detrimental to attaining the Heisenberg scaling in quantum metrology.
We study the interplay of control and parameter estimation on a quantum spin chain. A single qubit probe is attached to one end of the chain, while we wish to estimate a parameter on the other end. We find that control on the probe qubit can substantially improve the estimation performance and discover some interesting connections to quantum state transfer.
We study the role of quantum entanglement (particle entanglement and mode entanglement) in optical phase estimation by employing the first and second quantization formalisms of quantum mechanics. The quantum Fisher information (QFI) is expressed as a function of the first and second order optical coherence functions. The resulting form of the QFI elucidates the deriving metrological resources for quantum phase estimation: field intensity and photon detection correlations. In addition, our analysis confirms that mode entanglement is not required for quantum-enhanced interferometry, whereas part…
Optomechanically controlled response amplification for enhanced quantum sensing
Joint Laboratory of Optics of Palacký University and Institute of Physics ASCR, Faculty of Science, Palacký University（帕拉茨基大学与捷克科学院物理研究所联合光学实验室，帕拉茨基大学理学院）
Optical phase estimation is a vital measurement primitive that is used to perform accurate measurements of various physical quantities like length, velocity and displacements. The precision of such measurements can be largely enhanced by the use of entangled or squeezed states of light as demonstrated in a variety of different optical systems. Most of these accounts however deal with the measurement of a very small shift of an already known phase, which is in stark contrast to ab-initio phase estimation where the initial phase is unknown. Here we report on the realization of a quantum enhanced…
We report on the controllable response of the lasing frequency to the cavity round-trip path change. This is achieved by modifying the dispersion of the intracavity medium in the four-wave mixing regime in Rb. We can either increase the response by at least a factor of $2.7$ or drastically reduce it. The former regime is useful for sensitive measurements tracking the cavity round trip length and the latter regime is useful for precision metrology.
We study the simultaneous estimation of multiple phases as a discretised model for the imaging of a phase object. We identify quantum probe states that provide an enhancement compared to the best quantum scheme for the estimation of each individual phase separately, as well as improvements over classical strategies. Our strategy provides an advantage in the variance of the estimation over individual quantum estimation schemes that scales as O(d) where d is the number of phases. Finally, we study the attainability of this limit using realistic probes and photon-number-resolving detectors. This …
We present a general framework to study the simultaneous estimation of multiple phases in the presence of noise as a discretized model for phase imaging. This approach can lead to nontrivial bounds of the precision for multiphase estimation. Our results show that simultaneous estimation (SE) of multiple phases is always better than individual estimation (IE) of each phase even in noisy environment. However with $d$ being the number of phases, the $O(d)$ advantage in the variance of the estimation, with which SE outperforms IE schemes for noiseless processes, may disappear asymptotically. When …
We report on an orbital-angular-momentum-enhanced scheme for angular displacement estimation based on two-mode squeezed vacuum and parity detection. The sub-Heisenberg-limited sensitivity for angular displacement estimation is obtained in an ideal situation. Several realistic factors are also considered, including photon loss, dark counts, response-time delay, and thermal photon noise. Our results indicate that the effects of the realistic factors on the sensitivity can be offset by raising orbital angular momentum quantum number $\ell$. This reflects that the robustness and the practicability…
When standard light sources are employed, the precision of the phase determination is limited by the shot noise. Quantum entanglement provides means to exceed this limit with the celebrated example of N00N states that saturate the ultimate Heisenberg limit on precision, but at the same time are extremely fragile to losses. In contrast, we provide experimental evidence that appropriately engineered quantum states outperform both standard and N00N states in the precision of phase estimation when losses are present.
Quantum metrology is the state-of-the-art measurement technology. It uses quantum resources to enhance the sensitivity of phase estimation beyond what reachable within classical physics. While single parameter estimation theory has been widely investigated, much less is known about the simultaneous estimation of multiple phases, which finds key applications in imaging and sensing. In this manuscript we provide conditions of useful entanglement (among multimode particles, qudits) for multiphase estimation and adapt them to multiarm Mach-Zehnder interferometry. We discuss benchmark multimode Foc…
We propose a quantum metrology protocol for measuring frequencies and weak forces based on a periodic modulating quantum Jahn-Teller system composed of a single spin interacting with two bosonic modes. We show that in the first order of the frequency drive the time-independent effective Hamiltonian describes spin-dependent interaction between the two bosonic modes. In the limit of high-frequency drive and low bosonic frequency the quantum Jahn-Teller system exhibits critical behaviour which can be used for high-precision quantum estimation. A major advantage of our scheme is the robustness of …
I propose a quantum sensor based on driven-dissipative quantum system for the joint estimation of two conjugated variables characterizing the phase space displacement. The quantum probe consists of lattice system with two level atoms and bosonic modes which interact via dipolar coupling. Interplay between the coherent dynamics and dissipative processes of losses of bosonic excitations leads to a steady state which exhibits a non-analytical behaviour. I show that close to the dissipative phase transition the sensitivity of one of the conjugated parameters either the magnitude of the phase of th…
Quantum phase estimation protocols can provide a measuring method of phase shift with precision superior to standard quantum limit (SQL) due to the application of a nonclassical state of light. A squeezed vacuum state, whose variance in one quadrature is lower than the corresponding SQL, has been pointed out a sensitive resource for quantum phase estimation and the estimation accuracy is directly influenced by the properties of the squeezed state. Here we detailedly analyze the influence of the purity and squeezing level of the squeezed state on the accuracy of quantum phase estimation. The ma…
The angular displacement estimation is one of significant branches of quantum parameter estimation. However, most of the studies have focused on the single-angular displacement estimation, while the multiple angular displacement estimation in ideal and noisy scenarios is still elusive. In this paper, we investigate the simultaneous multiple angular displacement estimation based on an orbital angular momentum (OAM), together with inputting (d + 1)-mode NOON-like states as the probe state. By revealing the role of the intramode correlation of the probe state, this allows us to give a reasonable …
Precise parameter estimation plays a central role in science and technology. The statistical error in estimation can be decreased by repeating measurement, leading to that the resultant uncertainty of the estimated parameter is proportional to the square root of the number of repetitions in accordance with the central limit theorem. Quantum parameter estimation, an emerging field of quantum technology, aims to use quantum resources to yield higher statistical precision than classical approaches. Here, we report the first room-temperature implementation of entanglement-enhanced phase estimation…
For a fixed average energy, the simultaneous estimation of multiple phases can provide a better total precision than estimating them individually. We show this for a multimode interferometer with a phase in each mode, using Gaussian inputs and passive elements, by calculating the covariance matrix. The quantum Cramér-Rao bound provides a lower bound to the covariance matrix via the quantum Fisher information matrix, whose elements we derive to be the covariances of the photon numbers across the modes. We prove that this bound can be saturated. In spite of the Gaussian nature of the problem, th…
We analyze the performance of a generalized Kitaev's phase estimation algorithm where N phase gates, acting on $M$ qubits prepared in a product state, may be distributed in an arbitrary way. Unlike the standard algorithm, where the mean square error scales as 1/N, the optimal generalizations offer the Heisenberg $1/N^2$ error scaling and we show that they are in fact very close to the fundamental Bayesian estimation bound. We also demonstrate that the optimality of the algorithm breaks down when losses are taken into account, in which case the performance is inferior to the optimal entanglemen…
Momentum diffusion is a possible mechanism for driving macroscopic quantum systems towards classical behaviour. Experimental tests of this hypothesis rely on a precise estimation of the strength of this diffusion. We show that quantum-mechanical squeezing offers significant improvements, including when measuring position. For instance, with 10dB of mechanical squeezing, experiments would require a tenth of proposed free-fall times. Momentum measurement is better by an additional factor of three, while another quadrature is close to optimal. These have particular implications for the space-base…
Most studies in multiparameter estimation assume the dynamics is fixed and focus on identifying the optimal probe state and the optimal measurements. In practice, however, controls are usually available to alter the dynamics, which provides another degree of freedom. In this paper we employ optimal control methods, particularly the gradient ascent pulse engineering (GRAPE), to design optimal controls for the improvement of the precision limit in multiparameter estimation. We show that the controlled schemes not only capable to provide a higher precision limit, but also have a higher stability …
Optimizing quantum-enhanced Bayesian multiparameter estimation of phase and noise in practical sensors
Scuola Normale Superiore（高等师范学校）；Istituto Nanoscienze-CNR（意大利国家研究委员会纳米科学研究所）；Sapienza Università di Roma（罗马第一大学）；Universitá di Napoli “Federico II”（那不勒斯费德里科二世大学）；INFN, Sez. di Napoli（意大利国家核物理研究院那不勒斯分部）
The dynamics of quantum Fisher information (QFI) of the phase parameter in a driven two-state system is studied within the framework of non-Markovian dissipative process. The influences of memory effects, classical driving and detunings on the parameter-estimation precision are demonstrated by exactly solving the Hamiltonian under rotating-wave approximation. In sharp contrast with the results obtained in the presence of Markovian dissipation, we find that classical driving can drastically enhance the QFI, namely, the precision of parameter estimation in the non-Markovian regime. Moreover, the…
The advantage of quantum metrology has been experimentally demonstrated for phase estimations where the dynamics are commuting. General noncommuting dynamics, however, can have distinct features. For example, the direct sequential scheme, which can achieve the Heisenberg scaling for the phase estimation under commuting dynamics, can have even worse performances than the classical scheme under noncommuting dynamics. Here we realize a scalable optimally controlled sequential scheme, which can achieve the Heisenberg precision under general noncommuting dynamics. We also present an intuitive geome…
Squeezing-Enhanced Two-Phase Estimation with N-Particle W-type States
Weinan Normal University（渭南师范学院）；Engineering Research Center of X-ray Imaging and Detection, Universities of Shaanxi Province（陕西省X射线成像与探测工程研究中心）；School of Computer Science and Technology, Weinan Normal University（渭南师范学院计算机科学与技术学院）；Henan Normal University（河南师范大学）；Nanchang Hangkong University（南昌航空大学）
The quantum stochastic phase estimation has many applications in the precise measurement of various physical parameters. Similar to the estimation of a constant phase, there is a standard quantum limit for stochastic phase estimation, which can be obtained with the Mach-Zehnder interferometer and coherent input state. Recently, it has been shown that the stochastic standard quantum limit can be surpassed with non-classical resources such as the squeezed light. However, practical methods to achieve the quantum enhancement in the stochastic phase estimation remains largely unexplored. Here we pr…
We address the characterization of dissipative bosonic channels and show that estimation of the loss rate by Gaussian probes (coherent or squeezed) is improved in the presence of Kerr nonlinearity. In particular, enhancement of precision may be substantial for short interaction time, i.e. for media of moderate size, e.g. biological samples. We analyze in detail the behaviour of the quantum Fisher information (QFI), and determine the values of nonlinearity maximizing the QFI as a function of the interaction time and of the parameters of the input signal. We also discuss the precision achievable…
The first generation of multi-qubit quantum technologies will consist of noisy, intermediate-scale devices for which active error correction remains out of reach. To exploit such devices, it is thus imperative to use passive error protection that meets a careful trade-off between noise protection and resource overhead. Here, we experimentally demonstrate that single-qubit encoding can significantly enhance the robustness of entanglement and coherence of four-qubit graph states against local noise with a preferred direction. In particular, we explicitly show that local encoding provides a signi…
With the capability to find the best fit to arbitrarily complicated data patterns, machine-learning (ML) enhanced quantum state tomography (QST) has demonstrated its advantages in extracting complete information about the quantum states. Instead of using the reconstruction model in training a truncated density matrix, we develop a high-performance, lightweight, and easy-to-install supervised characteristic model by generating the target parameters directly. Such a characteristic model-based ML-QST can avoid the problem of dealing with large Hilbert space, but keep feature extractions with high…
Dephasing-Enhanced Response and Phase Transitions in Quantum Boltzmann Samplers
Nankai University（南开大学）；National University of Singapore（新加坡国立大学）；Chern Institute of Mathematics, Nankai University（南开大学陈省身数学研究所）；National Institute of Education, Nanyang Technological University（南洋理工大学国立教育学院）；Southwest Minzu University（西南民族大学）
We show that a significant quantum gain corresponding to squeezed or over-squeezed spin states can be obtained in multiparameter estimation by measuring the Hadamard coefficients of a 1D or 2D signal. The physical platform we consider consists of twolevel atoms in an optical lattice in a squeezed-Mott configuration, or more generally by correlated spins distributed in spatially separated modes. Our protocol requires the possibility to locally flip the spins, but relies on collective measurements. We give examples of applications to scalar or vector field mapping and compressed sensing.
Phase estimation algorithms are key protocols in quantum information processing. Besides applications in quantum computing, they can also be employed in metrology as they allow for fast extraction of information stored in the quantum state of a system. Here, we implement two suitably modified phase estimation procedures, the Kitaev- and the semiclassical Fourier-transform algorithms, using an artificial atom realized with a superconducting transmon circuit. We demonstrate that both algorithms yield a flux sensitivity exceeding the classical shot-noise limit of the device, allowing one to appro…
By exploiting the correlation properties of ultracold atoms in a multi-mode interferometer, we show how quantum enhanced measurement precision can be achieved with strong robustness to particle loss. While the potential for enhanced measurement precision is limited for even moderate loss in two-mode schemes, multi-mode schemes can be more robust. A ring interferometer for sensing rotational motion with non-interacting fermionic atoms can realize an uncertainty scaling of $1/(N\sqrtη)$ for $N$ particles with a fraction $η$ remaining after loss, which undercuts the shot noise limit of two mode i…
We propose a phase estimation protocol for optical interferometry that employs a probe state (containing on average n photons) obtained by squeezing each mode, separately, of a single photon path entangled Bell state. This scheme involves a Mach-Zehnder type interferometer for which each mode is squeezed after the first beam splitter. Information about the differential phase is extracted using a parity detection and the resulting measurement signal is super-resolving and supersensitive, with a minimum phase uncertainty 2/(n+1). This probe state can be generated with current technologies where …
The quantum enhanced classical sensor network consists of $K$ clusters of $N_e$ entangled quantum states that have been trialled $r$ times, each feeding into a classical estimation process. Previous literature has shown that each cluster can {ideally} achieve an estimation variance of $1/N_e^2r$ for sufficient $r$. We begin by deriving the optimal values for the minimum mean squared error of this quantum enhanced classical system. We then show that if noise is \emph{absent} in the classical estimation process, the mean estimation error will decay like $Ω(1/KN_e^2r)$. However, when noise is \em…
Quantum mechanics establishes the ultimate limit to the scaling of the precision on any parameter, by iden- tifying optimal probe states and measurements. While this paradigm is, at least in principle, adequate for the metrology of quantum channels involving the estimation of phase and loss parameters, we show that estimat- ing the loss parameters associated with a quantum channel and a realistic quantum detector are fundamentally different. While Fock states are provably optimal for the former, we identify a crossover in the nature of the optimal probe state for estimating detector imperfecti…
We study the generation of planar quantum squeezed (PQS) states by quantum non-demolition (QND) measurement of a cold ensemble of $^{87}$Rb atoms. Precise calibration of the QND measurement allows us to infer the conditional covariance matrix describing the $F_y$ and $F_z$ components of the PQS, revealing the dual squeezing characteristic of PQS. PQS states have been proposed for single-shot phase estimation without prior knowledge of the likely values of the phase. We show that for an arbitrary phase, the generated PQS gives a metrological advantage of at least 3.1 dB relative to classical st…
Sapienza Università di Roma（罗马第一大学）；Microsoft Research（微软研究院）
Phase estimation protocols provide a fundamental benchmark for the field of quantum metrology. The latter represents one of the most relevant applications of quantum theory, potentially enabling the capability of measuring unknown physical parameters with improved precision over classical strategies. Within this context, most theoretical and experimental studies have focused on determining the fundamental bounds and how to achieve them in the asymptotic regime where a large number of resources is employed. However, in most applications it is necessary to achieve optimal precisions by performin…
We consider quantum enhanced measurements with initially mixed states. We show very generally that for any linear propagation of the initial state that depends smoothly on the parameter to be estimated, the sensitivity is bound by the maximal sensitivity that can be achieved for any of the pure states from which the initial density matrix is mixed. This provides a very general proof that purely classical correlations cannot improve the sensitivity of parameter estimation schemes in quantum enhanced measurement schemes.
