Search

Using a random matrix approach, combined with the ergodicity hypothesis, we derive an analytical expression for the time-averaged quantum Fisher information (QFI) that applies to all quantum chaotic systems governed by Dyson's ensembles. Our approach integrates concepts of randomness, multipartite entanglement and quantum chaos. Furthermore, the QFI proves to be highly dependent on the dimension of the Krylov space confining the chaotic dynamics: it ranges from $N^2/3$ for $N$ qubits in the permutation-symmetric subspace (e.g. for chaotic kicked top models with long-range interactions), to $N$ when the dynamics extend over the full Hilbert space with or without bit reversal symmetry or parity symmetry (e.g. in chaotic models with short-range Ising-like interactions). In the former case, the QFI reveals multipartite entanglement among $N/3$ qubits. Interestingly this result can be related to isotropic substructures in the Wigner distribution of chaotic states and demonstrates the efficacy of quantum chaos for Heisenberg-scaling quantum metrology. Finally, our general expression for the QFI agrees with that obtained for random states and, differently from out-of-time-order-correlators, it can also distinguish chaotic from integrable unstable spin dynamics., Comment: 7+7 pages, 4+2 figures, comments are welcome

We study the multiparameter sensitivity bounds of a sensor made by an array of $d$ spatially-distributed Mach-Zehnder interferometers (MZIs). A generic single non-classical state is mixed with $d-1$ vacuums to create a $d$-modes entangled state, each mode entering one input port of a MZI, while a coherent state enters its second port. We show that local measurements, independently performed on each MZI, are sufficient to provide a sensitivity saturating the quantum Cram\'er-Rao bound. The sensor can overcome the shot noise limit for the estimation of arbitrary linear combinations of the $d$ phase shifts, provided that the non-classical probe state has an anti-squeezed quadrature variance. We compare the sensitivity bounds of this sensor with that achievable with $d$ independent MZIs, each probed with a nonclassical state and a coherent state. We find that the $d$ independent interferometers can achieve the same sensitivity of the entangled protocol but at the cost of using additional $d$ non-classical states rather than a single one. When using in the two protocols the same average number of particles per shot $\bar{n}_T$, we find analytically a sensitivity scaling $1/\bar{n}_T^2$ for the entangled case which provides a gain factor $d$ with respect to the separable case where the sensitivity scales as $d/\bar{n}_T^2$. We have numerical evidences that the gain factor $d$ is also obtained when fixing the total average number of particles, namely when optimizing with respect to the number of repeated measurements., Comment: 5 pages (3 figures) + Appendix (3 figures); comments are welcome

A central task in quantum metrology is to exploit quantum correlations to outperform classical sensitivity limits. Metrologically useful entanglement is identified when the quantum Fisher information (QFI) exceeds a separability bound for a given parameter-encoding Hamiltonian. However, so far, only results for linear Hamiltonians are well-established. Here, we characterize metrologically useful entanglement for nonlinear Hamiltonians, presenting separability bounds for collective angular momenta. Also, we provide a general expression for entangled states maximizing the QFI, showing that these are not always GHZ-like states. Finally, we compare the metrological usefulness of linear and nonlinear cases, in terms of entanglement detection and random symmetric states., Comment: 15 pages, 6 figures

Bounding the optimal precision in parameter estimation tasks is of central importance for technological applications. In the regime of a small number of measurements, or that of low signal-to-noise ratios, the meaning of common frequentist bounds such as the Cram\'er-Rao bound (CRB) become questionable. Here, we discuss different linear (Barankin-like) conditions that can be imposed on estimators and analyze when these conditions admit an optimal estimator with finite variance, for any number of measurement repetitions. We show that, if the number of imposed conditions is larger than the number of measurement outcomes, there generally does not exist a corresponding estimator with finite variance. We analyze this result from different viewpoints and examples and elaborate on connections to the shot-noise limit and the Kitaev phase estimation algorithm. We then derive an extended Cram\'er-Rao bound that is compatible with a finite variance in situations where the Barankin bound is undefined. Finally, we show an exemplary numerical confrontation between frequentist and Bayesian approaches to parameter estimation.

Recently, a new category of superfluids and superconductors has been discovered in various systems. These could be linked to the idea of a supersolid phase, featuring a macroscopic wavefunction with spatial modulation resulting from simultaneous, spontaneous breaking of gauge and translational symmetries. However, this relation has only been recognized in some cases and there is the need for universal properties quantifying the differences between supersolids and ordinary superfluids/superconductors or crystals. A key property is the superfluid fraction, which measures the reduction in superfluid stiffness due to spatial modulation, leading to the non-standard superfluid dynamics of supersolids. Here we employ the Josephson effect, common in superfluids and superconductors, to measure the superfluid fraction in a supersolid. Even without a physical barrier, the Josephson effect arises spontaneously in a supersolid due to spatial modulation. Individual lattice cells act as self-induced Josephson junctions, allowing the direct determination of the local superfluid fraction. We studied a cold-atom dipolar supersolid, revealing a significant sub-unity superfluid fraction. Our results open new research directions, enabling the exploration of novel phenomena like partially quantized vortices and supercurrents, potentially unifying the understanding of supersolid-like systems, and introducing a new type of Josephson junction., Comment: 15 pages, 10 figures

This document presents a summary of the 2023 Terrestrial Very-Long-Baseline Atom Interferometry Workshop hosted by CERN. The workshop brought together experts from around the world to discuss the exciting developments in large-scale atom interferometer (AI) prototypes and their potential for detecting ultralight dark matter and gravitational waves. The primary objective of the workshop was to lay the groundwork for an international TVLBAI proto-collaboration. This collaboration aims to unite researchers from different institutions to strategize and secure funding for terrestrial large-scale AI projects. The ultimate goal is to create a roadmap detailing the design and technology choices for one or more km-scale detectors, which will be operational in the mid-2030s. The key sections of this report present the physics case and technical challenges, together with a comprehensive overview of the discussions at the workshop together with the main conclusions., Comment: Summary of the Terrestrial Very-Long-Baseline Atom Interferometry Workshop held at CERN: https://indico.cern.ch/event/1208783/

We study the dynamical evolution of two quantum clocks interacting with a relativistic gravitational potential. We find a time dilation effect for the clocks in agreement with the gravitational time dilation as obtained from the Schwarzschild solution in General Relativity. We perform our investigation via the Page and Wootters quantum time formalism. The gravitational redshift, as emerging from our framework, is also proposed and discussed., Comment: 6 pages, 2 figures

We derive lower bounds on the variance of estimators in quantum metrology by choosing test observables that define constraints on the unbiasedness of the estimator. The quantum bounds are obtained by analytical optimization over all possible quantum measurements and estimators that satisfy the given constraints. We obtain hierarchies of increasingly tight bounds that include the quantum Cram\'er-Rao bound at the lowest order. In the opposite limit, the quantum Barankin bound is the variance of the locally best unbiased estimator in quantum metrology. Our results reveal generalizations of the quantum Fisher information that are able to avoid regularity conditions and identify threshold behavior in quantum measurements with mixed states, caused by finite data., Comment: 6+7 pages, 1+1 figures

Genuine multipartite nonlocality (GMN), the strongest form of multipartite nonlocality that describes fully collective nonlocal correlations among all experimental parties, can be observed when different distant parties each locally measure a particle from a shared entangled many-particle state. For the demonstration of GMN, the experimentally observed statistics are typically postselected: Events for which some parties do not detect a particle must be discarded. This coincidence postselection generally leads to the detection loophole that invalidates a proper nonlocality demonstration. In this work, we address how to close the detection loophole for a coincidence detection in demonstrations of nonlocality and GMN. We first show that if the number of detected particles is conserved, i.e., using ideal and noiseless experimental devices, one can employ causal diagrams and the no-signalling principle to prove that a coincidence postselection cannot create any detection loophole. Furthermore, for realistic experimental devices with finite detection efficiencies, we show how a general Bell inequality can be sharpened such that its new version is still valid after a postselection of the measurement data. In this case, there are threshold detection efficiencies that, if surpassed in the experiment, lead to the possibility to demonstrate nonlocality and GMN without opening the detection loophole. Our results imply that genuine $N$-partite nonlocality can be generated from $N$ independent particle sources even when allowing for non-ideal detectors.

Discarding undesirable measurement results in Bell experiments opens the detection loophole that prevents a conclusive demonstration of nonlocality. As closing the detection loophole represents a major technical challenge for many practical Bell experiments, it is customary to assume the so-called fair sampling assumption (FSA) that, in its original form, states that the collectively postselected statistics are a fair sample of the ideal statistics. Here, we analyze the FSA from the viewpoint of causal inference: We derive a causal structure that must be present in any causal model that faithfully encapsulates the FSA. This provides an easy, intuitive, and unifying approach that includes different accepted forms of the FSA and underlines what is really assumed when using the FSA. We then show that the FSA can not only be applied in scenarios with non-ideal detectors or transmission losses, but also in ideal experiments where only parts of the correlations are postselected, e.g., when the particles' destinations are in a superposition state. Finally, we demonstrate that the FSA is also applicable in multipartite scenarios that test for (genuine) multipartite nonlocality.

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.

We consider a global quantum system (the "Universe") satisfying a double constraint, both on total energy and total momentum. Generalizing the Page and Wootters quantum clock formalism, we provide a model of 3+1 dimensional, non-relativistic, quantum spacetime emerging from entanglement among different subsystems in a globally "timeless" and "positionless" Universe., Comment: Final version accepted for publication on AVS Quantum Science

Thanks to common-mode noise rejection, differential configurations are crucial for realistic applications of phase and frequency estimation with atom interferometers. Currently, differential protocols with uncorrelated particles and mode-separable settings reach a sensitivity bounded by the standard quantum limit (SQL). Here we show that differential interferometry can be understood as a distributed multiparameter estimation problem and can benefit from both mode and particle entanglement. Our protocol uses a single spin-squeezed state that is mode-swapped among common interferometric modes. The mode swapping is optimized to estimate the differential phase shift with sub-SQL sensitivity. Numerical calculations are supported by analytical approximations that guide the optimization of the protocol. The scheme is also tested with simulation of noise in atomic clocks and interferometers., Comment: 10 pages, 5 figures

We summarize the discussions at a virtual Community Workshop on Cold Atoms in Space concerning the status of cold atom technologies, the prospective scientific and societal opportunities offered by their deployment in space, and the developments needed before cold atoms could be operated in space. The cold atom technologies discussed include atomic clocks, quantum gravimeters and accelerometers, and atom interferometers. Prospective applications include metrology, geodesy and measurement of terrestrial mass change due to, e.g., climate change, and fundamental science experiments such as tests of the equivalence principle, searches for dark matter, measurements of gravitational waves and tests of quantum mechanics. We review the current status of cold atom technologies and outline the requirements for their space qualification, including the development paths and the corresponding technical milestones, and identifying possible pathfinder missions to pave the way for missions to exploit the full potential of cold atoms in space. Finally, we present a first draft of a possible road-map for achieving these goals, that we propose for discussion by the interested cold atom, Earth Observation, fundamental physics and other prospective scientific user communities, together with ESA and national space and research funding agencies., Comment: Summary of the Community Workshop on Cold Atoms in Space and corresponding Road-map: https://indico.cern.ch/event/1064855/

We consider a quantum Universe composed by a small system S and a large environment. It has been demonstrated that, for the vast majority of randomly chosen wave-functions of the Universe satisfying a total energy constraint, the reduced density matrix of the system S is given by the canonical statistical distribution. Here, through the Page and Wootters mechanism, we show that time and non-equilibrium dynamics can emerge as a consequence of the entanglement between the system and the environment present in the (randomly chosen) global wave-function of the Universe. The clock is provided by the environment, which ticks the temporal evolution of S. The paradox of the peaceful coexistence of statistical equilibrium and non-equilibrium dynamics is solved by identifying the trace over the environment degrees of freedom with the temporal trace over the entire history of the system S., Comment: Final version accepted for publication on Phys. Rev. D

We study a sensor network of distributed Mach-Zehnder interferometers (MZIs) for the parallel (simultaneous) estimation of an arbitrary number $d \geq 1$ of phase shifts. The scheme uses a squeezed-vacuum state that is split between $d$ modes by a quantum circuit (QC). Each output mode of the QC is the input of one of $d$ MZIs, the other input of each MZI being a coherent state. In particular, ${\it i}$) we identify the optimal configuration of the sensor network that allows the estimation of any linear combination of the $d$ phases with maximal sensitivity. The protocol overcomes the shot-noise limit and reaches Heisenberg scalings with respect to the total average number of particles in the overall probe state, the multiphase estimation only requiring local photocounting. Furthermore, the parallel multiphase estimation overcomes optimal separable strategies for the estimation of any linear combination of the phases: the sensitivity gain being a factor $d$, at most. Viceversa, ${\it ii}$) given a specific QC, we identify the optimal linear combination of the phases that maximizes the sensitivity and show that results are robust against random choices of the QC. Our scheme paves the ways to a variety of applications in distributed quantum sensing., Comment: 12 pages, 8 figures plus Appendix; comments are welcome

We discuss under which conditions multipartite entanglement in mixed quantum states can be characterized only in terms of two-point connected correlation functions, as it is the case for pure states. In turn, the latter correlations are defined via a suitable combination of (disconnected) one- and two-point correlation functions. In contrast to the case of pure states, conditions to be satisfied turn out to be rather severe. However, we were able to identify some interesting cases, as when the point-independence is valid of the one-point correlations in each possible decomposition of the density matrix, or when the operators that enter in the correlations are (semi-)positive/negative defined.

Recently there has been a great deal of interest surrounding the calibration of quantum sensors using machine learning techniques. In this work, we explore the use of regression to infer a machine-learned point estimate of an unknown parameter. Although the analysis is neccessarily frequentist - relying on repeated esitmates to build up statistics - we clarify that this machine-learned estimator converges to the Bayesian maximum a-posterori estimator (subject to some regularity conditions). When the number of training measurements are large, this is identical to the well-known maximum-likelihood estimator (MLE), and using this fact, we argue that the Cram{\'e}r-Rao sensitivity bound applies to the mean-square error cost function and can therefore be used to select optimal model and training parameters. We show that the machine-learned estimator inherits the desirable asymptotic properties of the MLE, up to a limit imposed by the resolution of the training grid. Furthermore, we investigate the role of quantum noise the training process, and show that this noise imposes a fundamental limit on number of grid points. This manuscript paves the way for machine-learning to assist the calibration of quantum sensors, thereby allowing maximum-likelihood inference to play a more prominent role in the design and operation of the next generation of ultra-precise sensors.

Improving the clock stability is of fundamental importance for the development of quantum-enhanced metrology. One of the main limitations arises from the randomly-fluctuating local oscillator (LO) frequency, which introduces "phase slips" for long interrogation times and hence failure of the frequency-feedback loop. Here we propose a strategy to improve the stability of atomic clocks by interrogating two out-of-phase state sharing the same LO. While standard Ramsey interrogation can only determine phases unambiguously in the interval $[-\pi/2,\pi/2]$, the joint interrogation allows for an extension to $[-\pi,\pi]$, resulting in a relaxed restriction of the Ramsey time and improvement of absolute clock stability. Theoretical predictions are supported by ab-initio numerical simulation for white and correlated LO noise. While our basic protocol uses uncorrelated atoms, we have further extended it to include spin-squeezing and further improving the scaling of clock stability with the number of atoms. Our protocol can be readily tested in current state-of-the-art experiments., Comment: 15 pages, 10 figures

We propose to use ultracold fermionic atoms in one-dimensional optical lattices to quantum simulate the electronic transport in quantum cascade laser (QCL) structures. The competition between the coherent tunneling among (and within) the wells and the dissipative decay at the basis of lasing is discussed. In order to validate the proposed simulation scheme, we quantitatively address such competition in a simplified one-dimensional model. We show the existence of optimal relationships between the model parameters, maximizing the particle current, the population inversion (or their product), and the stimulated emission rate. This substantiates the concept of emulating the QCL operation mechanisms in cold-atom optical lattice simulators, laying the groundwork for addressing open questions, such as the impact of electron-electron scattering and the origin of transport-induced noise, in the design of new-generation QCLs., Comment: 22 pages, 12 figures

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-signalling condition imposed by relativity. Finally, we apply our results to show that genuine three-partite nonlocality can be created with independent particle sources.

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 to implement a non-linear 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., Comment: 4 pages, 4 figues

We study the particle-entanglement dynamics witnessed by the quantum Fisher information (QFI) of a trapped Bose-Einstein condensate governed by the kicked rotor Hamiltonian. The dynamics is investigated with a beyond mean-field approach. We link the time scales of the validity of this approximation in, both, classical regular and chaotic regions, with the maximum Lyapunov exponents of the classical system. This establishes an effective connection between the classical chaos and the QFI. We finally study the critical point of a quantum phase transition using the beyond mean-field approximation by considering a two-mode bosonic Josephson junction with attractive interparticle interaction.

We propose a scheme for trapped atom interferometry using an interacting Bose-Einstein condensate. The condensate is controlled and spatially split in two confined external momentum modes through a series Bragg pulses. The proposed scheme (i) allows the generation of large entanglement in a trapped-interferometer configuration via one-axis twisting dynamic induced by interatomic interaction, and (ii) avoids the suppression of interactions during the interferometer sequence by a careful manipulation of the state before and after phase encoding. The interferometer can be used for the measurement of gravity with a sensitivity beyond the standard quantum limit., Comment: 5 pages, 4 figures

We characterize metrologically useful multipartite entanglement by representing partitions with Young diagrams. We derive entanglement witnesses that are sensitive to the shape of Young diagrams and show that Dyson's rank acts as a resource for quantum metrology. Common quantifiers, such as the entanglement depth and $k$-separability are contained in this approach as the diagram's width and height. Our methods are experimentally accessible in a wide range of atomic systems, as we illustrate by analyzing published data on the quantum Fisher information and spin-squeezing coefficients., Comment: 6 + 7 pages, 5 + 8 figures

Excited-state quantum phase transitions (ESQPTs) extend the notion of quantum phase transitions beyond the ground state. They are characterized by closing energy gaps amid the spectrum. Identifying order parameters for ESQPTs poses however a major challenge. We introduce spinor Bose-Einstein condensates as a versatile platform for studies of ESQPTs. Based on the mean-field dynamics, we define a topological order parameter that distinguishes between excited-state phases, and discuss how to interferometrically access the order parameter in current experiments. Our work opens the way for the experimental characterization of excited-state quantum phases in atomic many-body systems., Comment: 14 pages, 3 figures

Quantum phase estimation (QPE) is the key subroutine of several quantum computing algorithms as well as a central ingredient in quantum computational chemistry and quantum simulation. While QPE strategies have focused on the estimation of a single phase, applications to the simultaneous estimation of several phases may bring substantial advantages; for instance, in the presence of spatial or temporal constraints. In this work, we study a Bayesian algorithm for the parallel (simultaneous) estimation of multiple arbitrary phases. The protocol gives access to correlations in the Bayesian multi-phase distribution resulting in covariance matrix elements scaling inversely proportional to the square of the total number of quantum resources. The parallel estimation allows to surpass the sensitivity of sequential single-phase estimation strategies for optimal linear combinations of phases. Furthermore, the algorithm proves a certain noise resilience and can be implemented using single photons and standard optical elements in currently accessible experiments.

Quantum phase estimation (QPE) is one of the most important subroutines in quantum computing. In general applications, current QPE algorithms either suffer an exponential time overload or require a set of - notoriously quite fragile - GHZ states. These limitations have prevented so far the demonstration of QPE beyond proof-of-principles. Here we propose a new QPE algorithm that scales linearly with time and is implemented with a cascade of Gaussian spin states (GSS). GSS are renownedly resilient and have been created experimentally in a variety of platforms, from hundreds of ions up to millions of cold/ultracold neutral atoms. We show that our protocol achieves a QPE sensitivity overcoming previous bounds, including those obtained with GHZ states, and is robustly resistant to several sources of noise and decoherence. Our work paves the way toward realistic quantum advantage demonstrations of the QPE, as well as applications of atomic squeezed states for quantum computation., Comment: 9 pages, 3 figures, supplementary information not included, comment are welcome

We periodically modulate the lattice trapping potential of a $^{87}$Sr optical clock to Floquet engineer the clock transition. In the context of atomic gases in lattices, Floquet engineering has been used to shape the dispersion and topology of Bloch quasi-energy bands. Differently from these previous works manipulating the external (spatial) quasi-energies, we target the internal atomic degrees of freedom. We shape Floquet spin quasi-energies and measure their resonance profiles with Rabi spectroscopy. We provide the spectroscopic sensitivity of each band by measuring the Fisher information and show that this is not depleted by the Floquet dynamical modulation. The demonstration that the internal degrees of freedom can be selectively engineered by manipulating the external degrees of freedom inaugurates a novel device with potential applications in metrology, sensing and quantum simulations., Comment: 6 pages, 4 figures, including supplementary material, comments are welcome and more information at http://cqutp.org/users/xfzhang/

Bayesian estimation is a powerful theoretical paradigm for the operation of quantum sensors. However, the Bayesian method for statistical inference generally suffers from demanding calibration requirements that have so far restricted its use to proof-of-principle experiments. In this theoretical study, we formulate parameter estimation as a classification task and use artificial neural networks to efficiently perform Bayesian estimation. We show that the network's posterior distribution is centered at the true (unknown) value of the parameter within an uncertainty given by the inverse Fisher information, representing the ultimate sensitivity limit for the given apparatus. When only a limited number of calibration measurements are available, our machine-learning based procedure outperforms standard calibration methods. Thus, our work paves the way for Bayesian quantum sensors which can benefit from efficient optimization methods, such as in adaptive schemes, and take advantage of complex non-classical states. These capabilities can significantly enhance the sensitivity of future devices.

We propose a hybrid quantum-classical atomic clock protocol where the interrogation of an ensemble of uncorrelated atoms in a spin-coherent state is used to feedback one (or more) spin-squeezed atomic ensembles toward their optimal phase sensitivity point. This protocol overcomes the stability of a single Ramsey clock and it reaches a Heisenberg-limited stability while avoiding non-destructive measurements. Analytical predictions are compared with numerical simulations of clocks operations including correlated 1/f local oscillator noise., Comment: 6 pages, 3 figures

Time in quantum mechanics is peculiar: it is an observable that cannot be associated to an Hermitian operator. As a consequence it is impossible to explain dynamics in an isolated system without invoking an external classical clock, a fact that becomes particularly problematic in the context of quantum gravity. An unconventional solution was pioneered by Page and Wootters (PaW) in 1983. PaW showed that dynamics can be an emergent property of the entanglement between two subsystems of a static Universe. In this work we first investigate the possibility to introduce in this framework a Hermitian time operator complement of a clock Hamiltonian having an equally-spaced energy spectrum. An Hermitian operator complement of such Hamiltonian was introduced by Pegg in 1998, who named it "Age". We show here that Age, when introduced in the PaW context, can be interpreted as a proper Hermitian time operator conjugate to a "good" clock Hamiltonian. We therefore show that, still following Pegg's formalism, it is possible to introduce in the PaW framework bounded clock Hamiltonians with an unequally-spaced energy spectrum with rational energy ratios. In this case time is described by a POVM and we demonstrate that Pegg's POVM states provide a consistent dynamical evolution of the system even if they are not orthogonal, and therefore partially un-distinguishables., Comment: 13 pages

Quantum coherence is a crucial resource for quantum information processing. By employing the language of coherence orders largely applied in NMR systems, quantum coherence has been currently addressed in terms of multiple quantum coherences (MQCs). Here we investigate the $\alpha$-MQCs, a novel class of multiple quantum coherences which is based on $\alpha$-relative purity, an information-theoretic quantifier analogous to quantum fidelity and closely related to R\'{e}nyi relative entropy of order $\alpha$. Our framework enables linking $\alpha$-MQCs to Wigner-Yanase-Dyson skew information (WYDSI), an asymmetry monotone finding applications in quantum thermodynamics and quantum metrology. Furthermore, we derive a family of bounds on $\alpha$-MQCs, particularly showing that $\alpha$-MQC define a lower bound to quantum Fisher information (QFI). We illustrate these ideas for quantum systems described by single-qubit states, two-qubit Bell-diagonal states, and a wide class of multiparticle mixed states. Finally, we investigate the time evolution of the $\alpha$-MQC spectrum and the overall signal of relative purity, by simulating the time reversal dynamics of a many-body all-to-all Ising Hamiltonian and comment on applications to physical platforms such as NMR systems, trapped ions, and ultracold atoms., Comment: 19 pages, 9 figures. Close to published version

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., Comment: 6 pages, 2 figures

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., Comment: 8 + 10 pages, 4 + 1 figures

In a three-particle extension of Wheeler's delayed choice gedanken experiment, the quantum statistics of two particles is undetermined until a third particle is measured. As a function of the measurement result, the particles behave either as bosons or as fermions. The particles are distinguishable if no measurement is performed at all or when the measurement is performed in a rotated basis. The scheme is based on Greenberger-Horne-Zeilinger quantum correlations. It can be interpreted more generally as the encryption of maximally entangled states in a larger quantum superposition. The local quantum information is scrambled but can be decoded by the measurement result of a control particle. This can be extended to multiple particles and allows to develop quantum information protocols whose successful implementation depends on the collaboration of all parties., Comment: 9 pages, 3 figures

A closed-trajectory evolution of a quantum state generally imprints a phase that contains both dynamical and geometrical contributions. While dynamical phases depend on the reference system, geometric phase factors are uniquely defined by the properties of the outlined trajectory. Here, we generate and measure geometric phases in a Bose-Einstein condensate of $^{87}$Rb using a combination of dynamical quantum Zeno effect and measurement-free evolution. We show that the dynamical quantum Zeno effect can inhibit the formation of a geometric phase without altering the dynamical phase. This can be used to extract the geometric Aharonov-Anandan phase from any closed-trajectory evolution without requiring knowledge or control of the Hamiltonian., Comment: 4+3 pages, 4+3 figures

We show that matter-wave interference fringes formed by two overlapping atomic clouds can yield information about the non-local Bell correlations. To this end, we consider a simple atomic interferometer, where the clouds are released from the double-well potential and the relative phase is estimated from the density fit to this interference pattern. The Bell correlations can be deduced from the sensitivity of the phase obtained in this way. We examine the relation between these two quantities for a wide range of ground states of the double-well, scanning through the attractive and the repulsive interactions. The presented analysis includes the effects of finite temperature, when excited states are thermally occupied. We also consider the impact of the spatial resolution of the single-atom detectors and the fluctuations of the energy mismatch between the wells. These results establish a link between the fundamental (non-locality) and application-oriented (quantum metrology) aspects of entanglement., Comment: 5 pages, 2 figures

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 high-precision moment-based estimation of a phase parameter and can be used to systematically construct multipartite entanglement and non-classicality witnesses for complex quantum states., Comment: 6+2 pages, 1 figure

Non-vanishing fluctuations of the vacuum state are a salient feature of quantum theory. These fluctuations fundamentally limit the precision of quantum sensors. Nowadays, several systems such as optical clocks, gravitational wave detectors, matter-wave interferometers, magnetometers, and optomechanical systems approach measurement sensitivities where the effect of quantum fluctuations sets a fundamental limit, the so-called standard quantum limit (SQL). It has been proposed that the SQL can be overcome by squeezing the vacuum fluctuations. Realizations of this scheme have been demonstrated in various systems. However, protocols based on squeezed vacuum crucially rely on precise control of the relative orientation of the squeezing with respect to the operation imprinting the measured quantity. Lack of control can lead to an amplification of noise and reduces the sensitivity of the device. Here, we experimentally demonstrate a novel quantum metrological paradigm based on phase insensitive Fock states of the motional state of a trapped ion, with applications in frequency metrology and displacement detection. The measurement apparatus is used in two different experimental settings probing non-commuting observables with sensitivities beyond the SQL. In both measurements, classical preparation and detection noise are sufficiently small to preserve the quantum gain in a full metrological protocol.

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., Comment: 6 pages, 1 figure + Supplementary Material

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., Comment: 16 pages, 10 figures, close to the published version, Scientific Reports 8, 15663 (2018)

Frequentist and Bayesian phase estimation strategies lead to conceptually different results on the state of knowledge about the true value of the phase shift. 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 Cram\`er-Rao bound, for instance, do not apply to Bayesian strategies and vice-versa. Similarly, bounds for fluctuating parameters make no statement about the estimation of a fixed parameter., Comment: 4 figures

Understanding the distribution of quantum entanglement over many parties is a fundamental challenge of quantum physics and is of practical relevance for several applications in the field of quantum information. Here we use methods from quantum metrology to microscopically characterize the entanglement structure of multimode continuous-variable states in all possible multi-partitions and in all reduced distributions. From experimentally measured covariance matrices of Gaussian states with 2, 3, and 4 photonic modes with controllable losses, we extract the metrological sensitivity as well as an upper separability bound for each partition. An entanglement witness is constructed by comparing the two quantities. Our analysis demonstrates the usefulness of these methods for continuous-variable systems and provides a detailed geometric understanding of the robustness of cluster-state entanglement under photon losses.

The controlled generation and identification of quantum correlations, usually encoded in either qubits or continuous degrees of freedom, builds the foundation of quantum information science. Recently, more sophisticated approaches, involving a combination of two distinct degrees of freedom have been proposed to improve on the traditional strategies. Hyperentanglement describes simultaneous entanglement in more than one distinct degree of freedom, whereas hybrid entanglement refers to entanglement shared between a discrete and a continuous degree of freedom. In this work we propose a scheme that allows to combine the two approaches, and to extend them to the strongest form of quantum correlations. Specifically, we show how two identical, initially separated particles can be manipulated to produce Bell nonlocality among their spins, among their momenta, as well as across their spins and momenta. We discuss possible experimental realizations with atomic and photonic systems., Comment: 6 pages, 3 figures

We analyze families of measures for the quantum statistical speed which include as special cases the quantum Fisher information, the trace speed, i.e., the quantum statistical speed obtained from the trace distance, and more general quantifiers obtained from the family of Schatten norms. These measures quantify the statistical speed under generic quantum evolutions and are obtained by maximizing classical measures over all possible quantum measurements. We discuss general properties, optimal measurements, and upper bounds on the speed of separable states. We further provide a physical interpretation for the trace speed by linking it to an analog of the quantum Cram\'{e}r-Rao bound for median-unbiased quantum phase estimation., Comment: 19 pages, 1 figure