Tomography of a Number-Resolving Detector by Reconstruction of an Atomic Many-Body Quantum State.

The high-fidelity analysis of many-body quantum states of indistinguishable atoms requires the accurate counting of atoms. Here we report the tomographic reconstruction of an atom-number-resolving detector. The tomography is performed with an ultracold rubidium ensemble that is prepared in a coherent spin state by driving a Rabi coupling between the two hyperfine clock levels. The coupling is followed by counting the occupation number in one level. We characterize the fidelity of our detector and show that a negative-valued Wigner function is associated with it. Our results offer an exciting perspective for the high-fidelity reconstruction of entangled states and can be applied for a future demonstration of Heisenberg-limited atom interferometry.

Delta-Kick Squeezing.

We explore the possibility to overcome the standard quantum limit (SQL) in a free-fall atom interferometer using a Bose-Einstein condensate (BEC) in either of the two relevant cases of Bragg or Raman scattering light pulses. The generation of entanglement in the BEC is dramatically enhanced by amplifying the atom-atom interactions via the rapid action of an external trap, focusing the matter waves to significantly increase the atomic densities during a preparation stage-a technique we refer to as delta-kick squeezing (DKS). The action of a second DKS operation at the end of the interferometry sequence allows one to implement a nonlinear readout scheme, making the sub-SQL sensitivity highly robust against imperfect atom counting detection. We predict more than 30 dB of sensitivity gain beyond the SQL for the variance, assuming realistic parameters and 10^{6} atoms.

Genuine Multipartite Nonlocality with Causal-Diagram Postselection.

The generation and verification of genuine multipartite nonlocality (GMN) is of central interest for both fundamental research and quantum technological applications, such as quantum privacy. To demonstrate GMN in measurement data, the statistics are commonly postselected by neglecting undesired data. Until now, valid postselection strategies have been restricted to local postselection. A general postselection that is decided after communication between parties can mimic nonlocality, even though the complete data are local. Here, we establish conditions under which GMN is demonstrable even if observations are postselected collectively. Intriguingly, certain postselection strategies that require communication among several parties still offer a demonstration of GMN shared between all parties. The results are derived using the causal structure of the experiment and the no-signaling condition imposed by relativity. Finally, we apply our results to show that genuine three-partite nonlocality can be created with independent particle sources.

Quantifying computational advantage of Grover's algorithm with the trace speed.

Despite intensive research, the physical origin of the speed-up offered by quantum algorithms remains mysterious. No general physical quantity, like, for instance, entanglement, can be singled out as the essential useful resource. Here we report a close connection between the trace speed and the quantum speed-up in Grover's search algorithm implemented with pure and pseudo-pure states. For a noiseless algorithm, we find a one-to-one correspondence between the quantum speed-up and the polarization of the pseudo-pure state, which can be connected to a wide class of quantum statistical speeds. For time-dependent partial depolarization and for interrupted Grover searches, the speed-up is specifically bounded by the maximal trace speed that occurs during the algorithm operations. Our results quantify the quantum speed-up with a physical resource that is experimentally measurable and related to multipartite entanglement and quantum coherence.

Heisenberg-Limited Noisy Atomic Clock Using a Hybrid Coherent and Squeezed State Protocol.

We propose a hybrid quantum-classical atomic clock where the interrogation of atoms prepared in a spin-coherent (or weakly squeezed) state is used to feed back one or more highly spin-squeezed atomic states toward their optimal phase-sensitivity point. The hybrid clock overcomes the stability of a single Ramsey clock using coherent or optimal spin-squeezed states and reaches a Heisenberg-limited stability while avoiding nondestructive measurements. When optimized with respect to the total number of particles, the protocol surpasses the state-of-the-art proposals that use Greenberger-Horne-Zeilinger or NOON states. We compare analytical predictions with numerical simulations of clock operations, including correlated 1/f local oscillator noise.

Multiparameter squeezing for optimal quantum enhancements in sensor networks.

Squeezing currently represents the leading strategy for quantum enhanced precision measurements of a single parameter in a variety of continuous- and discrete-variable settings and technological applications. However, many important physical problems including imaging and field sensing require the simultaneous measurement of multiple unknown parameters. The development of multiparameter quantum metrology is yet hindered by the intrinsic difficulty in finding saturable sensitivity bounds and feasible estimation strategies. Here, we derive the general operational concept of multiparameter squeezing, identifying metrologically useful states and optimal estimation strategies. When applied to spin- or continuous-variable systems, our results generalize widely-used spin- or quadrature-squeezing parameters. Multiparameter squeezing provides a practical and versatile concept that paves the way to the development of quantum-enhanced estimation of multiple phases, gradients, and fields, and for the efficient characterization of multimode quantum states in atomic and optical sensor networks.

Motional Fock states for quantum-enhanced amplitude and phase measurements with trapped ions.

The quantum noise of the vacuum limits the achievable sensitivity of quantum sensors. In non-classical measurement schemes the noise can be reduced to overcome this limitation. However, schemes based on squeezed or Schrödinger cat states require alignment of the relative phase between the measured interaction and the non-classical quantum state. Here we present two measurement schemes on a trapped ion prepared in a motional Fock state for displacement and frequency metrology that are insensitive to this phase. The achieved statistical uncertainty is below the standard quantum limit set by quantum vacuum fluctuations, enabling applications in spectroscopy and mass measurements.

Metrological Nonlinear Squeezing Parameter.

The well-known metrological linear squeezing parameters (such as quadrature or spin squeezing) efficiently quantify the sensitivity of Gaussian states. Yet, these parameters are insufficient to characterize the much wider class of highly sensitive non-Gaussian states. Here, we introduce a class of metrological nonlinear squeezing parameters obtained by analytical optimization of measurement observables among a given set of accessible (possibly nonlinear) operators. This allows for the metrological characterization of non-Gaussian quantum states of discrete and continuous variables. Our results lead to optimized and experimentally feasible recipes for a high-precision moment-based estimation of a phase parameter and can be used to systematically construct multipartite entanglement and nonclassicality witnesses for complex quantum states.

Sensitivity Bounds for Multiparameter Quantum Metrology.

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 that saturate the precision bounds are provided.

Multipartite Entanglement at Finite Temperature.

The interplay of quantum and thermal fluctuations in the vicinity of a quantum critical point characterizes the physics of strongly correlated systems. Here we investigate this interplay from a quantum information perspective presenting the universal phase diagram of the quantum Fisher information at a quantum phase transition. Different regions in the diagram are identified by characteristic scaling laws of the quantum Fisher information with respect to temperature. This feature has immediate consequences on the thermal robustness of quantum coherence and multipartite entanglement. We support the theoretical predictions with the analysis of paradigmatic spin systems showing symmetry-breaking quantum phase transitions and free-fermion models characterized by topological phases. In particular we show that topological systems are characterized by the survival of large multipartite entanglement, reaching the Heisenberg limit at finite temperature.

Frequentist and Bayesian Quantum Phase Estimation.

Frequentist and Bayesian phase estimation strategies lead to conceptually different results on the state of knowledge about the true value of an unknown parameter. We compare the two frameworks and their sensitivity bounds to the estimation of an interferometric phase shift limited by quantum noise, considering both the cases of a fixed and a fluctuating parameter. We point out that frequentist precision bounds, such as the Cramér-Rao bound, for instance, do not apply to Bayesian strategies and vice versa. In particular, we show that the Bayesian variance can overcome the frequentist Cramér-Rao bound, which appears to be a paradoxical result if the conceptual difference between the two approaches are overlooked. Similarly, bounds for fluctuating parameters make no statement about the estimation of a fixed parameter.

Multipartite Entanglement in Topological Quantum Phases.

We witness multipartite entanglement in the ground state of the Kitaev chain-a benchmark model of a one dimensional topological superconductor-also with variable-range pairing, using the quantum Fisher information. Phases having a finite winding number, for both short- and long-range pairing, are characterized by a power-law diverging finite-size scaling of multipartite entanglement. Moreover, the occurring quantum phase transitions are sharply marked by the divergence of the derivative of the quantum Fisher information, even in the absence of a closing energy gap.

Optimal Measurements for Simultaneous Quantum Estimation of Multiple Phases.

A quantum theory of multiphase estimation is crucial for quantum-enhanced sensing and imaging and may link quantum metrology to more complex quantum computation and communication protocols. In this Letter, we tackle one of the key difficulties of multiphase estimation: obtaining a measurement which saturates the fundamental sensitivity bounds. We derive necessary and sufficient conditions for projective measurements acting on pure states to saturate the ultimate theoretical bound on precision given by the quantum Fisher information matrix. We apply our theory to the specific example of interferometric phase estimation using photon number measurements, a convenient choice in the laboratory. Our results thus introduce concepts and methods relevant to the future theoretical and experimental development of multiparameter estimation.

Witnessing entanglement without entanglement witness operators.

Quantum mechanics predicts the existence of correlations between composite systems that, although puzzling to our physical intuition, enable technologies not accessible in a classical world. Notwithstanding, there is still no efficient general method to theoretically quantify and experimentally detect entanglement of many qubits. Here we propose to detect entanglement by measuring the statistical response of a quantum system to an arbitrary nonlocal parametric evolution. We witness entanglement without relying on the tomographic reconstruction of the quantum state, or the realization of witness operators. The protocol requires two collective settings for any number of parties and is robust against noise and decoherence occurring after the implementation of the parametric transformation. To illustrate its user friendliness we demonstrate multipartite entanglement in different experiments with ions and photons by analyzing published data on fidelity visibilities and variances of collective observables.

Quantum-enhanced multiparameter estimation in multiarm interferometers.

Quantum metrology is the state-of-the-art measurement technology. It uses quantum resources to enhance the sensitivity of phase estimation over that achievable by 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 particle (qudit) entanglement for multiphase estimation and adapt them to multiarm Mach-Zehnder interferometry. We theoretically discuss benchmark multimode Fock states containing useful qudit entanglement and overcoming the sensitivity of separable qudit states in three and four arm Mach-Zehnder-like interferometers - currently within the reach of integrated photonics technology.

Spin-Mixing Interferometry with Bose-Einstein Condensates.

Unstable spinor Bose-Einstein condensates are ideal candidates to create nonlinear three-mode interferometers. Our analysis goes beyond the standard SU(1,1) parametric approach and therefore provides the regime of parameters where sub-shot-noise sensitivities can be reached with respect to the input total average number of particles. Decoherence due to particle losses and finite detection efficiency are also considered.

Quantum metrology. Fisher information and entanglement of non-Gaussian spin states.

Entanglement is the key quantum resource for improving measurement sensitivity beyond classical limits. However, the production of entanglement in mesoscopic atomic systems has been limited to squeezed states, described by Gaussian statistics. Here, we report on the creation and characterization of non-Gaussian many-body entangled states. We develop a general method to extract the Fisher information, which reveals that the quantum dynamics of a classically unstable system creates quantum states that are not spin squeezed but nevertheless entangled. The extracted Fisher information quantifies metrologically useful entanglement, which we confirm by Bayesian phase estimation with sub-shot-noise sensitivity. These methods are scalable to large particle numbers and applicable directly to other quantum systems.

Ultrasensitive two-mode interferometry with single-mode number squeezing.

A major challenge of the phase estimation problem is the engineering of high-intensity entangled probe states. The goal is to significantly enhance above the shot-noise limit the sensitivity of two-mode interferometers. Here we show that this can be achieved by squeezing in input, and then measuring in output, the population fluctuations of a single mode. The second input mode can be left as an arbitrary nonvacuum (e.g., a bright coherent) state. This two-mode state belongs to a novel class of particle-entangled states which are not spin squeezed. Already a 2.4 db gain above shot noise can be obtained when just a single-particle Fock state is injected into the empty input port of a classical interferometer configuration. Higher gains, up to the Heisenberg limit, can be reached with squeezed states of a larger number of particles. We finally study the robustness of this protocol with respect to detection noise.

Useful multiparticle entanglement and sub-shot-noise sensitivity in experimental phase estimation.

We experimentally demonstrate a general criterion to identify entangled states useful for the estimation of an unknown phase shift with a sensitivity higher than the shot-noise limit. We show how to exploit this entanglement on the examples of a maximum likelihood as well as of a Bayesian phase estimation protocol. Using an entangled four-photon state we achieve a phase sensitivity clearly beyond the shot-noise limit. Our detailed comparison of methods and quantum states for entanglement enhanced metrology reveals the connection between multiparticle entanglement and sub-shot-noise uncertainty, both in a frequentist and in a Bayesian phase estimation setting.

Localized and extended states in a disordered trap.

We study Anderson localization in a disordered potential combined with an inhomogeneous trap. We show that the spectrum displays both localized and extended states, which coexist at intermediate energies. In the region of coexistence, we find that the extended states result from confinement by the trap and are weakly affected by the disorder. Conversely, the localized states correspond to eigenstates of the disordered potential, which are only affected by the trap via an inhomogeneous energy shift. These results are relevant to disordered quantum gases and we propose a realistic scheme to observe the coexistence of localized and extended states in these systems.