Quantum Precision Measurement Based On Non-Gaussian Quantum States: An Introductory Review

In quantum metrology, entanglement represents a valuable resource that can be used to overcome the Standard Quantum Limit (SQL) that bounds the precision of sensors that operate with independent particles. Measurements beyond the SQL are typically enabled by relatively simple entangled states (squeezed states with Gaussian probability distributions), where quantum noise is redistributed between different quadratures. However, due to both fundamental limitations and the finite measurement resolution achieved in practice, sensors based on squeezed states typically operate far from the true fundamental limit of quantum metrology, the Heisenberg Limit. Here, by implementing an effective time-reversal protocol through a controlled sign change in an optically engineered many-body Hamiltonian, we demonstrate atomic-sensor performance with non-Gaussian states beyond the limitations of spin squeezing, and without the requirement of extreme measurement resolution. Using a system of 350 neutral $^{171}$Yb atoms, this signal amplification through time-reversed interaction (SATIN) protocol achieves the largest sensitivity improvement beyond the SQL ($11.8 \pm 0.5$~dB) demonstrated in any interferometer to date. Furthermore, we demonstrate a precision improving in proportion to the particle number (Heisenberg scaling), at fixed distance of 12.6~dB from the Heisenberg Limit. These results pave the way for quantum metrology using complex entangled states, with potential broad impact in science and technology. Potential applications include searches for dark matter and for physics beyond the standard model, tests of the fundamental laws of physics, timekeeping, and geodesy.

Non-Gaussian states with Wigner negativity are of particular interest in quantum technology due to their potential applications in quantum computing and quantum metrology. However, how to create such states at a remote location remains a challenge, which is important for efficiently distributing quantum resource between distant nodes in a network. Here, we experimentally prepare an optical non-Gaussian state with negative Wigner function at a remote node via local non-Gaussian operation and shared Gaussian entangled state existing quantum steering. By performing photon subtraction on one mode, Wigner negativity is created in the remote target mode. We show that the Wigner negativity is sensitive to loss on the target mode, but robust to loss on the mode performing photon subtraction. This experiment confirms the connection between the remotely created Wigner negativity and quantum steering. As an application, we present that the generated non-Gaussian state exhibits metrological power in quantum phase estimation.

Gaussian states have played an important role in the physics of continuous-variable quantum systems. They are appealing for the experimental ease with which they can be produced, and for their compact and elegant mathematical description. Nevertheless, many proposed quantum technologies require us to go beyond the realm of Gaussian states and introduce non-Gaussian elements. In this Tutorial, we provide a roadmap for the physics of non-Gaussian quantum states. We introduce the phase-space representations as a framework to describe the different properties of quantum states in continuous-variable systems. We then use this framework in various ways to explore the structure of the state space. We explain how non-Gaussian states can be characterized not only through the negative values of their Wigner function, but also via other properties such as quantum non-Gaussianity and the related stellar rank. For multimode systems, we are naturally confronted with the question of how non-Gaussian properties behave with respect to quantum correlations. To answer this question, we first show how non-Gaussian states can be created by performing measurements on a subset of modes in a Gaussian state. Then, we highlight that these measured modes must be correlated via specific quantum correlations to the remainder of the system to create quantum non-Gaussian or Wigner-negative states. On the other hand, non-Gaussian operations are also shown to enhance or even create quantum correlations. Finally, we demonstrate that Wigner negativity is a requirement to violate Bell inequalities and to achieve a quantum computational advantage. At the end of the Tutorial, we also provide an overview of several experimental realizations of non-Gaussian quantum states in quantum optics and beyond.

Quantum metrology uses quantum entanglement--correlations in the properties of microscopic systems--to improve the statistical precision of physical measurements. When measuring a signal, such as the phase shift of a light beam or an atomic state, a prominent limitation to achievable precision arises from the noise associated with the counting of uncorrelated probe particles. This noise, commonly referred to as shot noise or projection noise, gives rise to the standard quantum limit (SQL) to phase resolution. However, it can be mitigated down to the fundamental Heisenberg limit by entangling the probe particles. Despite considerable experimental progress in a variety of physical systems, a question that persists is whether these methods can achieve performance levels that compare favourably with optimized conventional (non-entangled) systems. Here we demonstrate an approach that achieves unprecedented levels of metrological improvement using half a million (87)Rb atoms in their 'clock' states. The ensemble is 20.1 ± 0.3 decibels (100-fold) spin-squeezed via an optical-cavity-based measurement. We directly resolve small microwave-induced rotations 18.5 ± 0.3 decibels (70-fold) beyond the SQL. The single-shot phase resolution of 147 microradians achieved by the apparatus is better than that achieved by the best engineered cold atom sensors despite lower atom numbers. We infer entanglement of more than 680 ± 35 particles in the atomic ensemble. Applications include atomic clocks, inertial sensors, and fundamental physics experiments such as tests of general relativity or searches for electron electric dipole moment. To this end, we demonstrate an atomic clock measurement with a quantum enhancement of 10.5 ± 0.3 decibels (11-fold), limited by the phase noise of our microwave source.

Phase estimation plays a central role in communications, sensing, and information processing. Quantum correlated states, such as squeezed states, enable phase estimation beyond the shot-noise limit, and in principle approach the ultimate quantum limit in precision, when paired with optimal quantum measurements. However, physical realizations of optimal quantum measurements for optical phase estimation with quantum-correlated states are still unknown. Here we address this problem by introducing an adaptive Gaussian measurement strategy for optical phase estimation with squeezed vacuum states that, by construction, approaches the quantum limit in precision. This strategy builds from a comprehensive set of locally optimal POVMs through rotations and homodyne measurements and uses the Adaptive Quantum State Estimation framework for optimizing the adaptive measurement process, which, under certain regularity conditions, guarantees asymptotic optimality for this quantum parameter estimation problem. As a result, the adaptive phase estimation strategy based on locally-optimal homodyne measurements achieves the quantum limit within the phase interval of [0,π/2). Furthermore, we generalize this strategy by including heterodyne measurements, enabling phase estimation across the full range of phases from [0,π), where squeezed vacuum allows for unambiguous phase encoding. Remarkably, for this phase interval, which is the maximum range of phases that can be encoded in squeezed vacuum, this estimation strategy maintains an asymptotic quantum-optimal performance, representing a significant advancement in quantum metrology.

We analyze general laws of continuous-variable entanglement dynamics during the deterministic attenuation and amplification of the physical signal carrying the entanglement. These processes are inevitably accompanied by noises, so we find fundamental limitations on noise intensities that destroy entanglement of gaussian and non-gaussian input states. The phase-insensitive amplification $\Phi_1 \otimes \Phi_2 \otimes \ldots \Phi_N$ with the power gain $\kappa_i \ge 2$ ($\approx 3$ dB, $i=1,\ldots,N$) is shown to destroy entanglement of any $N$-mode gaussian state even in the case of quantum limited performance. In contrast, we demonstrate non-gaussian states with the energy of a few photons such that their entanglement survives within a wide range of noises beyond quantum limited performance for any degree of attenuation or gain. We detect entanglement preservation properties of the channel $\Phi_1 \otimes \Phi_2$, where each mode is deterministically attenuated or amplified. Gaussian states of high energy are shown to be robust to very asymmetric attenuations, whereas non-gaussian states are at an advantage in the case of symmetric attenuation and general amplification. If $\Phi_1 = \Phi_2$, the total noise should not exceed $\frac{1}{2} \sqrt{\kappa^2+1}$ to guarantee entanglement preservation.

Multipartite entangled states are significant resources for both quantum information processing and quantum metrology. In particular, non-Gaussian entangled states are predicted to achieve a higher sensitivity of precision measurements than Gaussian states. On the basis of metrological sensitivity, the conventional linear Ramsey squeezing parameter (RSP) efficiently characterizes the Gaussian entangled atomic states but fails for much wider classes of highly sensitive non-Gaussian states. These complex non-Gaussian entangled states can be classified by the nonlinear squeezing parameter (NLSP), as a generalization of the RSP with respect to nonlinear observables and identified via the Fisher information. However, the NLSP has never been measured experimentally. Using a 19-qubit programmable superconducting processor, we report the characterization of multiparticle entangled states generated during its nonlinear dynamics. First, selecting ten qubits, we measure the RSP and the NLSP by single-shot readouts of collective spin operators in several different directions. Then, by extracting the Fisher information of the time-evolved state of all 19 qubits, we observe a large metrological gain of 9.89_{-0.29}^{+0.28} dB over the standard quantum limit, indicating a high level of multiparticle entanglement for quantum-enhanced phase sensitivity. Benefiting from high-fidelity full controls and addressable single-shot readouts, the superconducting processor with interconnected qubits provides an ideal platform for engineering and benchmarking non-Gaussian entangled states that are useful for quantum-enhanced metrology.

EPR steering is an asymmetric form of correlations which is intermediate between quantum entanglement and Bell nonlocality, and can be exploited for quantum communication with one untrusted party. In particular, steering of continuous variable Gaussian states has been extensively studied as a manifestation of the EPR paradox. While most of these studies focused on quadrature measurements for steering detection, two recent works revealed that there exist Gaussian states which are only steerable by non-Gaussian measurements. In this paper we perform a systematic investigation of EPR steering of bipartite Gaussian states by pseudospin measurements, complementing and extending previous findings. We first derive the density matrix elements of two-mode squeezed thermal states in the Fock basis, which may be of independent interest. We then use such a representation to investigate steering of these states as detected by a nonlinear criterion, based on second moments of the pseudospin correlation matrix. This analysis reveals previously unexplored regimes where non-Gaussian measurements are more effective than Gaussian ones to witness steering of Gaussian states in the presence of local noise. We further consider an alternative set of pseudospin observables, whose expectation value can be expressed compactly in terms of Wigner functions for all two-mode Gaussian states. However, according to the adopted criterion, these observables are found to be always less sensitive than Gaussian observables for steering detection. Finally, we investigate continuous variable Werner states, which are non-Gaussian mixtures of Gaussian states, and find that pseudospin measurements are always more effective than Gaussian ones to reveal their steerability. Our results provide useful insights on the role of non-Gaussian measurements in characterizing quantum correlations of Gaussian and non-Gaussian states.

Interferometry with quantum light is known to provide enhanced precision for\nestimating a single phase. However, depending on the parameters involved, the\nquantum limit for the simultaneous estimation of multiple parameters may not\nattainable, leading to trade-offs in the attainable precisions. Here we study\nthe simultaneous estimation of two parameters related to optical\ninterferometry: phase and loss, using a fixed number of photons. We derive a\ntrade-off in the estimation of these two parameters which shows that, in\ncontrast to single-parameter estimation, it is impossible to design a strategy\nsaturating the quantum Cramer-Rao bound for loss and phase estimation in a\nsingle setup simultaneously. We design optimal quantum states with a fixed\nnumber of photons achieving the best possible simultaneous precisions. Our\nresults reveal general features about concurrently estimating Hamiltonian and\ndissipative parameters, and has implications for sophisticated sensing\nscenarios such as quantum imaging.\n

Quantum metrology is the interdisciplinary of investigating how to utilize the principles of quantum mechanics to perform parameter estimation and improve the measurement precision by quantum effects. Attributed to well-developed techniques of quantum control, one can prepare several exotic non-Gaussian multi-particle entangled states in the ensembles of ultracold atoms. Based on many-body quantum interferometry, and using these non-Gaussian entangled states as probe, the high-precision measurement beyond the standard quantum limit can be realized . Utilizing suitable operations of quantum control, the non-Gaussian entangled states can be used for realizing Heisenberg-limited sensors, such as Sagnac interferometers, magnetometers, and many-body lock-in amplifiers. In this presentation, we will introduce the backgrounds and advances of this field.

Quantum metrology is the interdisciplinary of investigating how to utilize the principles of quantum mechanics to perform parameter estimation and improve the measurement precision by quantum effects. With the experimental developments of ultracold atoms, ultracold atomic ensemble provides an excellent platform for implementing quantum metrology. Attributed to well-developed techniques of quantum control, one can prepare several exotic non-Gaussian multi-particle entangled states in the ensembles of ultracold atoms. Based on many-body quanum interferometry, and using these non-Gaussian entangled states as probe, the high-precision measurement beyond the standard quantum limit can be realized. This article introduces the background and advancement of this field.

The measurement of physical quantities and measurement units standard promote the development of metrology. Especially, the developments of laser interference and atomic frequency standard bring a revolutionary leap for metrology. Many precision measurement techniques have been proposed and experimentally demonstrated, such as gravitational wave measurements and laser gyroscopes based on laser interferometry, and atomic clocks and atomic gyroscopes based on the atom interferometry. Recently, a new branch of science, quantum metrology, has grown up to further explore and exploit the quantum techniques for precision measurement of physical quantities.#br#This paper will focus on recent developments in quantum metrology and interference based on coherence and correlation of light and atom. Firstly, we briefly review the development of metrology. Then, we introduce our own researches in recent years, including quantum-correlation SU(1,1) optical interferometer based on four wave mixing process in atomic vapor and the atom-light hybrid interferometer based on Raman scattering in atomic vapor.#br#Interferometer is a powerful tool to measure physical quantities sensitive to the inference wave with high precision, and has been widely used in scientific research, industry test, navigation and guidance system. For example, the laser interferometer is able to measure optical phase sensitive quantities, including length, angular velocity, gravitational wave and so on. Meanwhile, the atom interferometer is sensitive to the change of atomic phase caused by the light, gravity, electric and magnetic fields. As a new type of interferometry, the atom-light hybrid interferometer, is sensitive to both the optical phase and atomic phase. Furthermore, SU(1,1) interferometer and nonlinear atom-light hybrid interferometer have the ability to beat the standard quantum limit of phase sensitivity. Quantum interference technology, whose phase measurement accuracy can break through the limit of standard quantum limit, is the core of quantum metrology and quantum measurement technology.

We introduce a new class of quantum many-particle entangled states, called the Dicke squeezed (or DS) states, which can be used to improve the precision in quantum metrology beyond the standard quantum limit. We show that the enhancement in measurement precision is characterized by a single experimentally detectable parameter, called the Dicke squeezing , which also bounds the entanglement depth for this class of states. The measurement precision approaches the ultimate Heisenberg limit as attains the minimum in an ideal Dicke state. Compared with other entangled states, we show that the DS states are more robust to decoherence and give better measurement precision under typical experimental noise.

Gaussian states and measurements collectively are not powerful-enough resources for quantum computing, as any Gaussian dynamics can be simulated efficiently, classically. However, it is known that any one non-Gaussian resource -- either a state, a unitary operation, or a measurement -- together with Gaussian unitaries, makes for universal quantum resources. Photon number resolving (PNR) detection, a readily-realizable non-Gaussian measurement, has been a popular tool to try and engineer non-Gaussian states for universal quantum processing. In this paper, we consider PNR detection of a subset of the modes of a zero-mean pure multi-mode Gaussian state as a means to herald a target non-Gaussian state on the undetected modes. This is motivated from the ease of scalable preparation of Gaussian states that have zero mean, using squeezed vacuum and passive linear optics. We calculate upper bounds on the fidelity between the actual heralded state and the target state. We find that this fidelity upper bound is $1/2$ when the target state is a multi-mode coherent cat-basis cluster state, a resource sufficient for universal quantum computing. This proves that there exist non-Gaussian states that are not producible by this method. Our fidelity upper bound is a simple expression that depends only on the target state represented in the photon-number basis, which could be applied to other non-Gaussian states of interest.

In quantum metrology, entangled states of many-particle systems are investigated to enhance measurement precision of the most precise clocks and field sensors. Whereas single-parameter quantum metrology is well established, joint multiparameter estimation poses conceptual challenges and has been explored only theoretically. We experimentally demonstrated multiparameter quantum metrology with an array of entangled atomic ensembles. By splitting a spin-squeezed ensemble, we created an atomic sensor array featuring intersensor entanglement that can be flexibly configured to enhance measurement precision of multiple parameters jointly. Using an optimal estimation protocol, we achieved substantial gains over the standard quantum limit in key multiparameter estimation tasks, thus grounding the concept of quantum enhancement of field sensor arrays and imaging devices.