A well-studied scenario in quantum parameter estimation theory arises when the parameter to be estimated is imprinted on the initial state by a Hamiltonian of the form $θG$. For such "phase shift Hamiltonians" it has been shown that one cannot improve the channel quantum Fisher information by adding ancillas and letting the system interact with them. Here we investigate the general case, where the Hamiltonian is not necessarily a phase shift, and show that in this case in general it \emph{is} possible to increase the quantum channel information and to reach an upper bound. This can be done by adding a term proportional to the derivative of the Hamiltonian, or by subtracting a term to the original Hamiltonian. Both methods do not make use of any ancillas and show therefore that for quantum channel estimation with arbitrary parameter-dependent Hamiltonian, entanglement with an ancillary system is not necessary to reach the best possible sensitivity. By adding an operator to the Hamiltonian we can also modify the time scaling of the channel quantum Fisher information. We illustrate our techniques with NV-center magnetometry and the estimation of the direction of a magnetic field in a given plane using a single spin-1 as probe.
We identify precision limits for the simultaneous estimation of multiple parameters in multimode interferometers. Quantum strategies to enhance the multiparameter sensitivity are based on entanglement among particles, modes or combining both. The maximum attainable sensitivity of particle-separable states defines the multiparameter shot-noise limit, which can be surpassed without mode entanglement. Further enhancements up to the multiparameter Heisenberg limit are possible by adding mode entanglement. Optimal strategies which saturate the precision bounds are provided.
We propose and analyze a new approach based on parity-time ($\mathcal{PT}$) symmetric microcavities with balanced gain and loss to enhance the performance of cavity-assisted metrology. We identify the conditions under which $\mathcal{PT}$-symmetric microcavities allow to improve sensitivity beyond what is achievable in loss-only systems. We discuss its application to the detection of mechanical motion, and show that the sensitivity is significantly enhanced in the vicinity of the transition point from unbroken- to broken-$\mathcal{PT}$ regimes. We believe that our results open a new direction …
We find a phase matching condition for enhancement of sensitivity in a Mach-Zehnder interferometer illuminated by an arbitrary state in one input port and an odd(even) state in the other port. Under this condition, the Fisher information becomes maximal with respect to the relative phase of two modes and the phase sensitivity is enhanced. For the case with photon losses, we further find that the phase matching condition keeps unchanged with a coherent state and a coherent superposition state as the input states.
Enhancing Quantum Metrology with High-order Fisher Information and Experiments
School of Information Science and Technology, Southwest Jiaotong University（西南交通大学信息科学与技术学院）；School of Mathematical Sciences, Capital Normal University（首都师范大学数学科学学院）；Hefei National Laboratory（合肥国家实验室）
Sensitivity Bounds of Multiparameter Metrology at Thermal Equilibrium
College of Electronics and Information Engineering, Tongji University（同济大学电子与信息工程学院）；Shanghai Research Institute for Intelligent Autonomous Systems, Tongji University（同济大学上海智能自主系统研究院）
The quantum Cramér-Rao bound is a cornerstone of modern quantum metrology, as it provides the ultimate precision in parameter estimation. In the multiparameter scenario, this bound becomes a matrix inequality, which can be cast to a scalar form with a properly chosen weight matrix. Multiparameter estimation thus elicits tradeoffs in the precision with which each parameter can be estimated. We show that, if the information is encoded in a unitary transformation, we can naturally choose the weight matrix as the metric tensor linked to the geometry of the underlying algebra $\mathfrak{su}(n)$, wi…
We show a general relationship between a superposition of macroscopically distinct states and sensitivity in quantum metrology. Generalized cat states are defined by using an index which extracts the coherence between macroscopically distinct states, and a wide variety of states, including a classical mixture of an exponentially large number of states, has been identified as the generalized cat state with this criterion. We find that, if we use the generalized cat states for magnetic field sensing without noise, we achieve the Heisenberg limited sensitivity. Moreover, we even show that sensiti…
In this paper we present a novel apparatus aimed at measuring very small birefringences and anisotropies, based on frequency metrology and not on polarimetry as usual. In our experiment, a very high finesse resonant cavity is used to convert the phase difference into a resonance frequency difference, which can then be measured with very high accuracy. We describe the set-up and present the results of experimental tests which exhibited a sensitivity dn ~ 2 x 10?18, allowing for the measurement of long-predicted magneto-electro-optical effects in gases. Since the shotnoise limited sensitivity of…
We study the sensitivity of a Mach-Zehnder interferometer that contains in addition to the phase shifter a non-linear element. By including both elements in a cavity or a loop that the light transverses many times, a non-linear kicked version of the interferometer arises. We study its sensitivity as function of the phase shift, the kicking strength, the maximally reached average number of photons, and damping due to photon loss for an initial coherent state. We find that for vanishing damping Heisenberg-limited scaling of the sensitivity arises if squeezing dominates the total photon number. F…
We propose and analyze a new approach based on quantum error correction (QEC) to improve quantum metrology in the presence of noise. We identify the conditions under which QEC allows one to improve the signal-to-noise ratio in quantum-limited measurements, and we demonstrate that it enables, in certain situations, Heisenberg-limited sensitivity. We discuss specific applications to nanoscale sensing using nitrogen-vacancy centers in diamond in which QEC can significantly improve the measurement sensitivity and bandwidth under realistic experimental conditions.
The goal of quantum metrology is the precise estimation of parameters using quantum properties such as entanglement. This estimation usually consists of three steps: state preparation, time evolution during which information of the parameters is encoded in the state, and readout of the state. Decoherence during the time evolution typically degrades the performance of quantum metrology and is considered to be one of the major obstacles to realizing entanglement-enhanced sensing. We show, however, that under suitable conditions, this decoherence can be exploited to improve the sensitivity. Assum…
Quantum metrology that employs weak-values can potentially effectuate parameter estimation with an ultra-high sensitivity and has been typically explored across quantum optics setups. Recognizing the importance of sensitive parameter estimation in the solid-state, we propose a spintronic device platform to realize this. The setup estimates a very weak localized Zeeman splitting by exploiting a resonant tunneling enhanced magnetoresistance readout. We establish that this paradigm offers nearly optimal performance with a quantum Fisher information enhancement of about $10^4$ times that of single…
Ramsey interferometry is routinely used in quantum metrology for the most sensitive measurements of optical clock frequencies. Spontaneous decay to the electromagnetic vacuum ultimately limits the interrogation time and thus sets a lower bound to the optimal frequency sensitivity. In dense ensembles of two-level systems the presence of collective effects such as superradiance and dipole-dipole interaction tends to decrease the sensitivity even further. We show that by a redesign of the Ramsey-pulse sequence to include different rotations of individual spins that effectively fold the collective…
We study the sensitivity and resolution of phase measurement in a Mach-Zehnder interferometer with two-mode squeezed vacuum (<n> photons on average). We show that super-resolution and sub-Heisenberg sensitivity is obtained with parity detection. In particular, in our setup, dependence of the signal on the phase evolves <n> times faster than in traditional schemes, and uncertainty in the phase estimation is better than 1/<n>.
We present several entanglement criteria in terms of the quantum Fisher information that help to relate various forms of multipartite entanglement to the sensitivity of phase estimation. We show that genuine multipartite entanglement is necessary to reach the maximum sensitivity in some very general metrological tasks using a two-arm linear interferometer. We also show that it is needed to reach the maximum average sensitivity in a certain combination of such metrological tasks.
We investigate the sensitivity of a recently proposed method for precision measurement [Phys. Rev. Lett. 106, 140502 (2011)], focusing on an implementation based on solid-state spin systems. The scheme amplifies a quantum sensor response to weak external fields by exploiting its coupling to spin impurities in the environment. We analyze the limits to the sensitivity due to decoherence and propose dynamical decoupling schemes to increase the spin coherence time. The sensitivity is also limited by the environment spin polarization; therefore we discuss strategies to polarize the environment spin…
We study the phase sensitivity in the conventional $SU(2)$ and nonconventional $SU(1,1)$ interferometers with the coherent and squeezed vacuum input state via the quantum Cramer-Rao bound. We explicitly construct the detection scheme that gives the optimal phase sensitivity. For practical purposes, we show that in the presence of photon loss, both interferometers with proper homodyne detections, are nearly optimal. We also find that unlike the coherent state and squeezed vacuum state, the effects of the imperfect detector on the phase sensitivity cannot be asymptotically removed for a generic …
Quantum metrology promises high-precision measurements of classical parameters with far reaching implications for science and technology. So far, research has concentrated almost exclusively on quantum-enhancements in integrable systems, such as precessing spins or harmonic oscillators prepared in non-classical states. Here we show that large benefits can be drawn from rendering integrable quantum sensors chaotic, both in terms of achievable sensitivity as well as robustness to noise, while avoiding the challenge of preparing and protecting large-scale entanglement. We apply the method to spin…
Quantum metrology promises improved sensitivity in parameter estimation over classical procedures. However, there is an extensive debate over the question how the sensitivity scales with the resources (such as the average photon number) and number of queries that are used in estimation procedures. Here, we reconcile the physical definition of the relevant resources used in parameter estimation with the information-theoretical scaling in terms of the query complexity of a quantum network. This leads to a completely general optimality proof of the Heisenberg limit for quantum metrology. We give …
Quantum metrology studies the use of entanglement and other quantum resources to improve precision measurement. An interferometer using N independent particles to measure a parameter X can achieve at best the "standard quantum limit" (SQL) of sensitivity δX \propto N^{-1/2}. The same interferometer using N entangled particles can achieve in principle the "Heisenberg limit" δX \propto N^{-1}, using exotic states. Recent theoretical work argues that interactions among particles may be a valuable resource for quantum metrology, allowing scaling beyond the Heisenberg limit. Specifically, a k-parti…
The Heisenberg limit is the superior precision available by entanglement sensors. However, entanglementis fragile against dephasing, and there is no known quantum metrology protocol that can achieve Heisenberg limited sensitivity with the presence of independent dephasing. Here, we show that the Heisenberg limit is attainable under the effect of independent dephasing under conditions where the probe qubits decohere due to both target fields and local environments. To detect the target fields, we exploit the entanglement properties to decay much faster than the classical states due to collectiv…
The impact of measurement imperfections on quantum metrology protocols has not been approached in a systematic manner so far. In this work, we tackle this issue by generalising firstly the notion of quantum Fisher information to account for noisy detection, and propose tractable methods allowing for its approximate evaluation. We then show that in canonical scenarios involving $N$ probes with local measurements undergoing readout noise, the optimal sensitivity depends crucially on the control operations allowed to counterbalance the measurement imperfections -- with global control operations, …
Quantum algorithms such as dynamical decoupling can be used to improve the sensitivity of a quantum sensor to a signal while suppressing sensitivity to noise. Atomic clocks are among the most sensitive quantum sensors, with recent improvements in clock technology allowing for unprecedented precision and accuracy. These clocks are highly sensitive to variations in fundamental constants, making them ideal probes for local ultralight scalar dark matter. Further improvements to the sensitivity is expected in proposed nuclear clocks based on the thorium 229m isomer. We investigate the use of variou…
Quantum metrology with entangled resources aims to achieve sensitivity beyond the standard quantum limit by harnessing quantum effects even in the presence of environmental noise. So far, sensitivity has been mainly discussed from the viewpoint of reducing statistical errors under the assumption of perfect knowledge of a noise model. However, we cannot always obtain complete information about a noise model due to coherence time fluctuations, which are frequently observed in experiments. Such unknown fluctuating noise leads to systematic errors and nullifies the quantum advantages. Here, we pro…
We establish the nonclassicality of continuous-variable states as a resource for quantum metrology. Based on the quantum Fisher information of multimode quadratures, we introduce the metrological power as a measure of nonclassicality with a concrete operational meaning of displacement sensitivity beyond the classical limit. This measure belongs to the resource theory of nonclassicality, which is nonincreasing under linear optical elements. Our Letter reveals that a single copy, highly nonclassical quantum state is intrinsically advantageous when compared to multiple copies of a quantum state w…
Sub-Planck phase-space structures in the Wigner function of the motional degree of freedom of a trapped ion can be used to perform weak force measurements with Heisenberg-limited sensitivity. We propose methods to engineer the Hamiltonian of the trapped ion to generate states with such small scale structures, and we show how to use them in quantum metrology applications.
I consider the role of detection noise in quantum-enhanced metrology in collective spin systems, and derive a fundamental bound for the maximum obtainable sensitivity for a given level of added detection noise. I then present an interaction-based readout utilising the commonly used one-axis twisting scheme that approaches this bound for states generated via several commonly considered methods of generating quantum enhancement, such as one-axis twisting, two-axis counter-twisting, twist-and-turn squeezing, quantum non-demolition measurements, and adiabatically scanning through a quantum phase t…
Recently, Motes et al. proposed in Phys. Rev. Lett. 114, 170802 (2015) a linear optics interferometer with N identical single photon input states as a tool for sub-shot-noise phase estimation which does not require NOON states sources. This thesis investigates how the degree of distinguishability between the input photonic spectra affects the interferometric phase sensitivity, by employing the results obtained by Tamma and Laibacher in Phys. Rev. Lett. 114, 243601 (2015). Remarkably, the obtained results show, that for small phases, the phase sensitivity in the case of completely distinguishab…
Including collisional decoherence explicitly, phase sensitivity for estimating effective scattering strength $χ$ of a two-component Bose-Einstein condensate is derived analytically. With a measurement of spin operator $\hat{J}_{x}$, we find that the optimal sensitivity depends on initial coherent spin state. It degrades by a factor of $(2γ)^{1/3}$ below super-Heisenberg limit $\propto 1/N^{3/2}$ for particle number $N$ and the dephasing rate $1<\!<γ<N^{3/4}$. With a $\hat{J}_y$ measurement, our analytical results confirm that the phase $ϕ=χt\sim 0$ can be detected at the limit even in the pres…
Quantum optical metrology aims to identify ultimate sensitivity bounds for the estimation of parameters encoded into quantum states of the electromagnetic field. In many practical applications, including imaging, microscopy, and remote sensing, the parameter of interest is not only encoded in the quantum state of the field, but also in its spatio-temporal distribution, i.e. in its mode structure. In this mode-encoded parameter estimation setting, we derive an analytical expression for the quantum Fisher information valid for arbitrary multimode Gaussian fields. To illustrate the power of our a…
We demonstrate that a many-body nonlocality is a resource for ultra-precise metrology. This result is achieved by linking the sensitivity of a quantum sensor with a combination of many-body correlation functions that witness the nonlocality. We illustrate our findings with some prominent examples---a collection of spins forming an Ising chain and a gas of ultra-cold atoms in any two-mode configuration.
The radical pair (RP) mechanism, which describes the quantum dynamics of a spatially separated electron pair, is considered as one of the principal models of avian magnetoreception. Different from the conventional phenomenological approach where the sensitivity of avian magnetoreception is characterized by the singlet yield $Φ_{S}$, we introduce the quantum Fisher information (QFI), which represents the maximum information about the magnetic field's direction extracted from the RP state, to give a precise measure of sensitivity of avian compass essentially. The consistency between our results …
We study very generally to what extent the uncertainty with which a phase shift can be estimated in quantum metrology can be reduced by extending the Hamiltonian that generates the phase shift to an ancilla system with a Hilbert space of arbitrary dimension, and allowing arbitrary interactions between the original system and the ancilla. Such Hamiltonian extensions provide a general framework for open quantum systems, as well as for "non-linear metrology schemes" that have been investigated over the last few years. We prove that such Hamiltonian extensions cannot improve the sensitivity of the…
Quantum-dense metrology (QDM) constitutes a special case of quantum metrology in which two orthogonal phase space projections of a signal are simultaneously sensed beyond the shot noise limit. Previously it was shown that the additional sensing channel that is provided by QDM contains information that can be used to identify and to discard corrupted segments from the measurement data. Here, we demonstrate a proof-of-principle experiment in which this information is used for improving the sensitivity without discarding any measurement segments. Our measurement reached sub-shot-noise performance…
Physical implementations of quantum information processing devices are generally not unique, and we are faced with the problem of choosing the best implementation. Here, we consider the sensitivity of quantum devices to variations in their different components. To measure this, we adopt a quantum metrological approach, and find that the sensitivity of a device to variations in a component has a particularly simple general form. We use the concept of cost functions to establish a general practical criterion to decide between two different physical implementations of the same quantum device cons…
Interferometric phase measurement is widely used to precisely determine quantities such as length, speed, and material properties. Without quantum correlations, the best phase sensitivity $Δφ$ achievable using $n$ photons is the shot noise limit (SNL), $Δφ=1/\sqrt{n}$. Quantum-enhanced metrology promises better sensitivity, but despite theoretical proposals stretching back decades, no measurement using photonic (i.e. definite photon number) quantum states has truly surpassed the SNL. Rather, all such demonstrations --- by discounting photon loss, detector inefficiency, or other imperfections -…
We introduce a super-sensitive phase measurement technique that yields the Heisenberg limit without using either a squeezed state or a many-particle entangled state. Instead, we use a many-particle separable quantum state to probe the phase and we then retrieve the phase through single-particle interference. The particles that physically probe the phase are never detected. Our scheme involves no coincidence measurement or many-particle interference and yet exhibits phase super-resolution. We also analyze in detail how the loss of probing particles affects the measurement sensitivity and find t…
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{…
Optimal observables for (non-)equilibrium quantum metrology from the master equation
Instituto Galego de Física de Altas Enerxías (IGFAE), Universidade de Santiago de Compostela（加利西亚高能物理研究所，圣地亚哥-德孔波斯特拉大学）；Department of Physics, Korea University（韩国大学物理系）
Nonclassical correlations beyond entanglement might provide a resource in quantum information tasks, such as quantum computation or quantum metrology. Quantum discord is a measure of nonclassical correlations, to which entanglement belongs as a subset. Exploring the operational meaning of quantum discord as a resource in quantum information processing tasks, such as quantum metrology, is of essential importance to our understanding of nonclassical correlations. In our recent work [Phys. Rev. A, 98, 012115 (2018)], we considered a protocol---which we call the greedy local thermometry protocol--…
Quantum metrology makes use of coherent superpositions to detect weak signals. While in principle the sensitivity can be improved by increasing the density of sensing particles, in practice this improvement is severely hindered by interactions between them. Using a dense ensemble of interacting electronic spins in diamond, we demonstrate a novel approach to quantum metrology. It is based on a new method of robust quantum control, which allows us to simultaneously eliminate the undesired effects associated with spin-spin interactions, disorder and control imperfections, enabling a five-fold enh…
Quantum metrology uses small changes in the output probabilities of a quantum measurement to estimate the magnitude of a weak interaction with the system. The sensitivity of this procedure depends on the relation between the input state, the measurement results, and the generator observable describing the effect of the weak interaction on the system. This is similar to the situation in weak measurements, where the weak value of an observable exhibits a symmetric dependence on initial and final conditions. In this paper, it is shown that the phase sensitivity of a quantum measurement is in fact…
We develop a general framework for parameter estimation that allows only trusted parties to access the result and achieves optimal precision. The protocols are designed such that adversaries can access some information indeterministically, but only at the risk of getting caught (cheat-sensitivity); under the assumption that the adversary can access the channel only once, then the protocol is unconditionally secure. By combining techniques from quantum cryptography and quantum metrology, we devise cryptographic procedures for single parameter estimation when an arbitrary number of parties are i…
We present experimental and theoretical results on a new interferometer topology that nests a SU(2) interferometer, e.g., a Mach-Zehnder or Michelson interferometer, inside a SU(1,1) interferometer, i.e., a Mach-Zehnder interferometer with parametric amplifiers in place of beam splitters. This SU(2)-in-SU(1,1) nested interferometer (SISNI) simultaneously achieves high signal-to-noise ratio (SNR), sensitivity beyond the standard quantum limit (SQL) and tolerance to photon losses external to the interferometer, e.g., in detectors. We implement a SISNI using parametric amplification by four-wave …
