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Universal control of multiple qubits -- the ability to entangle qubits and to perform arbitrary individual qubit operations -- is a fundamental resource for quantum computation, simulation, and networking. Here, we implement a new laser-free scheme for universal control of trapped ion qubits based on microwave magnetic fields and radiofrequency magnetic field gradients. We demonstrate high-fidelity entanglement and individual control by creating symmetric and antisymmetric two-qubit maximally entangled states with fidelities in the intervals [0.9983, 1] and [0.9964, 0.9988], respectively, at 68% confidence, corrected for state initialization error. This technique is robust against multiple sources of decoherence, usable with essentially any trapped ion species, and has the potential to perform simultaneous entangling operations on many pairs of ions without increasing control signal power or complexity.

The quantum statistical fluctuations of the electromagnetic field establish a limit, known as the shot-noise limit, on the sensitivity of optical measurements performed with classical technologies. However, quantum technologies are not constrained by this shot-noise limit. In this regard, the possibility of using every photon produced by quantum sources of light to estimate small physical parameters, beyond the shot-noise limit, constitutes one of the main goals of quantum optics. Here we experimentally demonstrate a scalable protocol for quantum-enhanced optical phase estimation across a broad range of phases, with neither pre- nor post-selected measurements. This is achieved through the efficient design of a source of spontaneous parametric down-conversion in combination with photon-number-resolving detection. The robustness of two-mode squeezed vacuum states against loss allows us to outperform schemes based on N00N states, in which the loss of a single photon is enough to remove all phase information from a quantum state. In contrast to other schemes that rely on N00N states or conditional measurements, the sensitivity of our technique could be improved through the generation and detection of high-order photon pairs. This unique feature of our protocol makes it scalable. Our work is important for quantum technologies that rely on multiphoton interference such as quantum imaging, boson sampling and quantum networks., Comment: 9 pages, 6 figures, added more discussions

Quantum entanglement of mechanical systems emerges when distinct objects move with such a high degree of correlation that they can no longer be described separately. Although quantum mechanics presumably applies to objects of all sizes, directly observing entanglement becomes challenging as masses increase, requiring measurement and control with a vanishingly small error. Here, using pulsed electromechanics, we deterministically entangle two mechanical drumheads with masses of 70 pg. Through nearly quantum-limited measurements of the position and momentum quadratures of both drums, we perform quantum state tomography and thereby directly observe entanglement. Such entangled macroscopic systems are uniquely poised to serve in fundamental tests of quantum mechanics, enable sensing beyond the standard quantum limit, and function as long-lived nodes of future quantum networks., Comment: 42 pages, 3 main figures, supplementary materials: text, 4 figures, 3 tables, 7 references

The Models and Modeling Working Group at PME-NA has provided a forum for discussing and collaborating on projects fundamental to research on mathematical modeling since the first PME-NA conference in 1978. We propose to convene this Working Group at PME-NA 41 with a dual purpose: (1) to build on a theme begun in Greenville, holding a focused workshop to pursue innovations in activity design to connect modeling activities to mathematically rich, socially-engaged inquiry into questions from the world outside of school; and (2) to continue to invite newcomers to the Models and Modeling Perspective (MMP), giving them an introduction to this design research tradition. [For the complete proceedings, see ED606556.]

Large-scale quantum computers will require quantum gate operations between widely separated qubits. A method for implementing such operations, known as quantum gate teleportation (QGT), requires only local operations, classical communication, and shared entanglement. We demonstrate QGT in a scalable architecture by deterministically teleporting a controlled-NOT (CNOT) gate between two qubits in spatially separated locations in an ion trap. The entanglement fidelity of our teleported CNOT is in the interval [0.845, 0.872] at the 95% confidence level. The implementation combines ion shuttling with individually-addressed single-qubit rotations and detections, same- and mixedspecies two-qubit gates, and real-time conditional operations, thereby demonstrating essential tools for scaling trapped-ion quantum computers combined in a single device.

Quantum state tomography aims to determine the quantum state of a system from measured data and is an essential tool for quantum information science. When dealing with continuous variable quantum states of light, tomography is often done by measuring the field amplitudes at different optical phases using homodyne detection. The quadrature-phase homodyne measurement outputs a continuous variable, so to reduce the computational cost of tomography, researchers often discretize the measurements. We show that this can be done without significantly degrading the fidelity between the estimated state and the true state. This paper studies different strategies for determining the histogram bin widths. We show that computation time can be significantly reduced with little loss in the fidelity of the estimated state when the measurement operators corresponding to each histogram bin are integrated over the bin width., Comment: 10 pages, 8 figures

A Bell test is a randomized trial that compares experimental observations against the philosophical worldview of local realism. A Bell test requires spatially distributed entanglement, fast and high-efficiency detection and unpredictable measurement settings. Although technology can satisfy the first two of these requirements, the use of physical devices to choose settings in a Bell test involves making assumptions about the physics that one aims to test. Bell himself noted this weakness in using physical setting choices and argued that human `free will' could be used rigorously to ensure unpredictability in Bell tests. Here we report a set of local-realism tests using human choices, which avoids assumptions about predictability in physics. We recruited about 100,000 human participants to play an online video game that incentivizes fast, sustained input of unpredictable selections and illustrates Bell-test methodology. The participants generated 97,347,490 binary choices, which were directed via a scalable web platform to 12 laboratories on five continents, where 13 experiments tested local realism using photons, single atoms, atomic ensembles, and superconducting devices. Over a 12-hour period on 30 November 2016, participants worldwide provided a sustained data flow of over 1,000 bits per second to the experiments, which used different human-generated data to choose each measurement setting. The observed correlations strongly contradict local realism and other realistic positions in bipartite and tripartite scenarios. Project outcomes include closing the `freedom-of-choice loophole' (the possibility that the setting choices are influenced by `hidden variables' to correlate with the particle properties), the utilization of video-game methods for rapid collection of human generated randomness, and the use of networking techniques for global participation in experimental science., Comment: This version includes minor changes resulting from reviewer and editorial input. Abstract shortened to fit within arXiv limits

Estimation of quantum states and measurements is crucial for the implementation of quantum information protocols. The standard method for each is quantum tomography. However, quantum tomography suffers from systematic errors caused by imperfect knowledge of the system. We present a procedure to simultaneously characterize quantum states and measurements that mitigates systematic errors by use of a single high-fidelity state preparation and a limited set of high-fidelity unitary operations. Such states and operations are typical of many state-of-the-art systems. For this situation we design a set of experiments and an optimization algorithm that alternates between maximizing the likelihood with respect to the states and measurements to produce estimates of each. In some cases, the procedure does not enable unique estimation of the states. For these cases, we show how one may identify a set of density matrices compatible with the measurements and use a semi-definite program to place bounds on the state's expectation values. We demonstrate the procedure on data from a simulated experiment with two trapped ions., Comment: updated to be almost identical to published version. 13 two-column pages, 5 figures

From dice to modern complex circuits, there have been many attempts to build increasingly better devices to generate random numbers. Today, randomness is fundamental to security and cryptographic systems, as well as safeguarding privacy. A key challenge with random number generators is that it is hard to ensure that their outputs are unpredictable. For a random number generator based on a physical process, such as a noisy classical system or an elementary quantum measurement, a detailed model describing the underlying physics is required to assert unpredictability. Such a model must make a number of assumptions that may not be valid, thereby compromising the integrity of the device. However, it is possible to exploit the phenomenon of quantum nonlocality with a loophole-free Bell test to build a random number generator that can produce output that is unpredictable to any adversary limited only by general physical principles. With recent technological developments, it is now possible to carry out such a loophole-free Bell test. Here we present certified randomness obtained from a photonic Bell experiment and extract 1024 random bits uniform to within $10^{-12}$. These random bits could not have been predicted within any physical theory that prohibits superluminal signaling and allows one to make independent measurement choices. To certify and quantify the randomness, we describe a new protocol that is optimized for apparatuses characterized by a low per-trial violation of Bell inequalities. We thus enlisted an experimental result that fundamentally challenges the notion of determinism to build a system that can increase trust in random sources. In the future, random number generators based on loophole-free Bell tests may play a role in increasing the security and trust of our cryptographic systems and infrastructure., Comment: Substantial overlap with arXiv:1702.05178. Compared to arXiv:1702.05178, the current manuscript reports new experimental results with experimental improvements described in a new Methods section. There are also some small adjustments to theory. A revised version of this manuscript has been accepted for publication in Nature

Random numbers are an important resource for applications such as numerical simulation and secure communication. However, it is difficult to certify whether a physical random number generator is truly unpredictable. Here, we exploit the phenomenon of quantum nonlocality in a loophole-free photonic Bell test experiment for the generation of randomness that cannot be predicted within any physical theory that allows one to make independent measurement choices and prohibits superluminal signaling. To certify and quantify the randomness, we describe a new protocol that performs well in an experimental regime characterized by low violation of Bell inequalities. Applying an extractor function to our data, we obtained 256 new random bits, uniform to within 0.001., Comment: 32 pages

We report correlation measurements on two $^9$Be$^+$ ions that violate a chained Bell inequality obeyed by any local-realistic theory. The correlations can be modeled as derived from a mixture of a local-realistic probabilistic distribution and a distribution that violates the inequality. A statistical framework is formulated to quantify the local-realistic fraction allowable in the observed distribution without the fair-sampling or independent-and-identical-distributions assumptions. We exclude models of our experiment whose local-realistic fraction is above 0.327 at the 95 \% confidence level. This bound is significantly lower than 0.586, the minimum fraction derived from a perfect Clauser-Horne-Shimony-Holt inequality experiment. Furthermore, our data provides a device-independent certification of the deterministically created Bell states.

Maximum likelihood quantum state tomography yields estimators that are consistent, provided that the likelihood model is correct, but the maximum likelihood estimators may have bias for any finite data set. The bias of an estimator is the difference between the expected value of the estimate and the true value of the parameter being estimated. This paper investigates bias in the widely used maximum likelihood quantum state tomography. Our goal is to understand how the amount of bias depends on factors such as the purity of the true state, the number of measurements performed, and the number of different bases in which the system is measured. For that, we perform numerical experiments that simulate optical homodyne tomography under various conditions, perform tomography, and estimate bias in the purity of the estimated state. We find that estimates of higher purity states exhibit considerable bias, such that the estimates have lower purity than the true states., Comment: New version where we fixed error in evenly spaced phases. Expanded analysis of evenly spaced phases. 8 figures. Submitted to Phys. Rev. A

We report high-fidelity laser-beam-induced quantum logic gates on magnetic-field-insensitive qubits comprised of hyperfine states in $^{9}$Be$^+$ ions with a memory coherence time of more than 1 s. We demonstrate single-qubit gates with error per gate of $3.8(1)\times 10^{-5}$. By creating a Bell state with a deterministic two-qubit gate, we deduce a gate error of $8(4)\times10^{-4}$. We characterize the errors in our implementation and discuss methods to further reduce imperfections towards values that are compatible with fault-tolerant processing at realistic overhead., Comment: 12 pages, 7 figures

Entangled states are a crucial resource for quantum-based technologies such as quantum computers and quantum communication systems (1,2). Exploring new methods for entanglement generation is important for diversifying and eventually improving current approaches. Here, we create entanglement in atomic ions by applying laser fields to constrain the evolution to a restricted number of states, in an approach that has become known as "quantum Zeno dynamics" (3-5). With two trapped $^9\rm{Be}^+$ ions, we obtain Bell state fidelities up to $0.990^{+2}_{-5}$, with three ions, a W-state (6) fidelity of $0.910^{+4}_{-7}$ is obtained. Compared to other methods of producing entanglement in trapped ions, this procedure is relatively insensitive to certain imperfections such as fluctuations in laser intensity, laser frequency, and ion-motion frequencies., Comment: 44 pages, 10 figures