The indistinguishability of many bosons undergoing passive linear transformations followed by number basis measurements is fully characterized by the visible state of the bosons. However, measuring all the parameters in the visible state is experimentally demanding. In this work, we seek to perform partial characterization of the visible state by measuring properties of it that are available after randomization. First, we study the case where the occupied visible modes are randomly permuted, and second, we study the case where Haar random linear optical unitaries are applied. In each case, we find that the generalized bunching probability-which is the probability that all the input bosons arrive in a given subset of the output modes-obeys monotonicity with respect to some partial order of distinguishability of the input bosons. As an intermediate result, we show that Lieb's permanental-dominance conjecture for immanants is equivalent to the following statement: for states that are invariant under permutations of the occupied visible modes, the generalized bunching probability is maximized when the bosons are perfectly indistinguishable. We also prove that a consequence of the monotonicity of the generalized bunching probability after Haar averaging is that this average is maximized when the bosons are perfectly indistinguishable. Finally, we discuss applications of our results to thermometry of cold-atom systems.

The unpredictability of random numbers is fundamental to both digital security1,2 and applications that fairly distribute resources3,4. However, existing random number generators have limitations-the generation processes cannot be fully traced, audited and certified to be unpredictable. The algorithmic steps used in pseudorandom number generators5 are auditable, but they cannot guarantee that their outputs were a priori unpredictable given knowledge of the initial seed. Device-independent quantum random number generators6-9 can ensure that the source of randomness was unknown beforehand, but the steps used to extract the randomness are vulnerable to tampering. Here we demonstrate a fully traceable random number generation protocol based on device-independent techniques. Our protocol extracts randomness from unpredictable non-local quantum correlations, and uses distributed intertwined hash chains to cryptographically trace and verify the extraction process. This protocol forms the basis for a public traceable and certifiable quantum randomness beacon that we have launched10. Over the first 40 days of operation, we completed the protocol 7,434 out of 7,454 attempts-a success rate of 99.7%. Each time the protocol succeeded, the beacon emitted a pulse of 512 bits of traceable randomness. The bits are certified to be uniform with error multiplied by actual success probability bounded by 2-64. The generation of certifiable and traceable randomness represents a public service that operates with an entanglement-derived advantage over comparable classical approaches.

We propose and implement a device-independent protocol for quantum position verification against unentangled adversaries. Our experiment achieves provable localization to a 1-dimensional region that is 40 . 7(7) % the size of the smallest theoretical region achievable with classical protocols.

As quantum error correction (QEC) experiments continue to make rapid progress, there is increased interest in designing experiments with guarantees of logical performance. At present, one difficulty is the lack of a clear connection between logical performance and the low-level error models. In this work, we take an important step toward addressing this issue by proving that consecutive QEC cycles of a stabilizer code with Pauli stochastic noise and with a single-cycle infidelity ε_1 ≤ 1/64 admit an approximate logical Markovian model, meaning that consecutive noisy QEC cycles can be modeled by a memoryless error process acting only on the logical subsystem. The approximate logical Markovian model can be computed from the low-level error model, and the deviations from the true behavior are exponentially suppressed in the number of QEC cycles. Consequently, we expect that the approximate logical Markovian model will be both a useful tool for logical characterization and an aid for designing stabilizer-code implementations with guarantees of logical performance.

A general one-dimensional quantum optical mode is described by a shape in the time or frequency domain. A fundamental problem is to measure a quadrature operator of such a mode. If the shape is narrow in frequency this can be done by pulsed homodyne detection, in which the mode and a matched local oscillator (LO) interfere on a beamsplitter, whose output ports are monitored by photo-detectors. The quadrature value is proportional to the difference between the photo-detectors' signals. When the shape of the mode is broad in frequency, the lack of uniform response of the detectors across the spectrum prevents direct application of this technique. We show that pulsed homodyne detection can be generalized to broadband pulsed (BBP) homodyne detection setups with detectors such as calorimeters that detect total energy instead of total number of photons. This generalization has applications in quantum measurements of femtosecond pulses, and, speculatively, measurements of Rindler modes to verify the temperature of Unruh radiation. Like pulsed homodyne detection, BBP homodyne detection requires choosing the LO pulse such that the subtracted signal approaches the desired quadrature measurement for large LO amplitudes. A distinctive feature of the technique is that the LO pulse does not belong to the mode of the quadrature being measured. We analyze how the implemented measurement approaches an ideal quadrature measurement with growing LO amplitude. We prove that the moments of the measurement converge to the moments of the quadrature and that the measurement distributions converge weakly.

Gaussian states are ubiquitous in quantum optics and information processing, and it is essential to have effective tools for their characterization. One such tool is a photon-number-resolving detector, and the simplest configuration involves counting the total number of photons in the state to be characterized. This motivates the following question: What properties of a multimode Gaussian state are determined by the signal from one detector that measures the total number of photons up to some bound? We find that if the Gaussian state occupies S modes and the probabilities of n photons for all n 8S are known, then we can determine the spectrum of the Gaussian covariance matrix and the magnitude of the displacements in each eigenspace of the covariance matrix. Nothing more can be learned, even if all photon-number probabilities are known. When the state is pure, the covariance matrix spectrum determines the squeezing parameters of the state.

We implement a traceable protocol for device-independent random number generation. Our demonstration involves transparently integrating a device-independent RNG into a public randomness beacon, and feeding 7434 random number pulses into the beacon over 39 days.

Precise quantum control and measurement of several harmonic oscillators, such as the modes of the electromagnetic field in a cavity or of mechanical motion, are key for their use as quantum platforms. The motional modes of trapped ions can be individually controlled and have good coherence properties. However, achieving high-fidelity two-mode operations and nondestructive measurements of the motional state has been challenging. Here we demonstrate the coherent exchange of single motional quanta between spectrally separated harmonic motional modes of a trapped-ion crystal. The timing, strength, and phase of the coupling are controlled through an oscillating electric potential with suitable spatial variation. Coupling rates that are much larger than decoherence rates enable demonstrations of high fidelity quantum state transfer and beamsplitter operations, entanglement of motional modes, and Hong-Ou-Mandel-type interference. Additionally, we use the motional coupling to enable repeated non-destructive projective measurement of a trapped-ion motional state. Our work enhances the suitability of trapped-ion motion for continuous-variable quantum computing and error correction and may provide opportunities to improve the performance of motional cooling and motion-mediated entangling interactions.

Consider minimizing the entropy of a mixture of states by choosing each state subject to constraints. If the spectrum of each state is fixed, we expect that in order to reduce the entropy of the mixture, we should make the states less distinguishable in some sense. Here, we study a class of optimization problems that are inspired by this situation and shed light on the relevant notions of distinguishability. The motivation for our study is the recently introduced spin alignment conjecture. In the original version of the underlying problem, each state in the mixture is constrained to be a freely chosen state on a subset of $n$ qubits tensored with a fixed state $Q$ on each of the qubits in the complement. According to the conjecture, the entropy of the mixture is minimized by choosing the freely chosen state in each term to be a tensor product of projectors onto a fixed maximal eigenvector of $Q$, which maximally "aligns" the terms in the mixture. We generalize this problem in several ways. First, instead of minimizing entropy, we consider maximizing arbitrary unitarily invariant convex functions such as Fan norms and Schatten norms. To formalize and generalize the conjectured required alignment, we define alignment as a preorder on tuples of self-adjoint operators that is induced by majorization. We prove the generalized conjecture for Schatten norms of integer order, for the case where the freely chosen states are constrained to be classical, and for the case where only two states contribute to the mixture and $Q$ is proportional to a projector. The last case fits into a more general situation where we give explicit conditions for maximal alignment. The spin alignment problem has a natural "dual" formulation, versions of which have further generalizations that we introduce.

A boson sampler implements a restricted model of quantum computing. It is defined by the ability to sample from the distribution resulting from the interference of identical bosons propagating according to programmable, non-interacting dynamics1. An efficient exact classical simulation of boson sampling is not believed to exist, which has motivated ground-breaking boson sampling experiments in photonics with increasingly many photons2–12. However, it is difficult to generate and reliably evolve specific numbers of photons with low loss, and thus probabilistic techniques for postselection7 or marked changes to standard boson sampling10–12 are generally used. Here, we address the above challenges by implementing boson sampling using ultracold atoms13,14 in a two-dimensional, tunnel-coupled optical lattice. This demonstration is enabled by a previously unrealized combination of tools involving high-fidelity optical cooling and imaging of atoms in a lattice, as well as programmable control of those atoms using optical tweezers. When extended to interacting systems, our work demonstrates the core abilities required to directly assemble ground and excited states in simulations of various Hubbard models15,16. Boson sampling using ultracold atoms in a two-dimensional, tunnel-coupled optical lattice is enabled by high-fidelity programmable control with optical tweezers of a large number of atoms trapped in an optical lattice.

The reliability of quantum resources can be compromised in practice due to the complexity of their generation processes and/or the potential manipulations by untrusted parties during transmission. When performing an information task with an unreliable quantum resource, it is incorrect to treat the random variables associated with repeated experimental trials as independent and identically distributed (i.i.d.). To certify the performance of such a task, one can make a random decision in each trial, either to spot-check some property of the quantum resource or to utilize the resource for the task. The task considered can be quantum key distribution, quantum randomness expansion, verifiable quantum computation, or resource allocation in quantum networks. Unfortunately, existing methods for certifying quantum performance through spot-checking are not suitable for non-i.i.d. repeated trials without additional assumptions. Here we present a novel method to address this challenge. The method works efficiently with a finite number of non-i.i.d. trials. Furthermore, our method can be adapted to estimate quantum properties in situations where the quantum resource is spot-checked and destroyed by a measurement during each non-i.i.d. repeated trial.

The continuing improvement in the qualities of photon-number-resolving (PNR) detectors opens new possibilities for measuring quantum states of light. In this work we consider the question of what properties of an arbitrary multi-mode Gaussian state are determined by a single PNR detector that measures total-photon number. We find an answer to this question in the ideal case where the exact photon-number probabilities are known. We show that the quantities determined by the total-photon-number distribution are the spectrum of the covariance matrix, the absolute displacement in each eigenspace of the covariance matrix, and nothing else. In the case of pure Gaussian states, the spectrum determines the squeezing parameters.

Randomized benchmarking (RB) is a widely used strategy to assess the quality of available quantum gates in a computational context. RB involves applying known random sequences of gates to an initial state and using the statistics of a final measurement step to determine an effective depolarizing error per step of the sequence, which is a metric of gate quality. Here we investigate the advantages of fully randomized benchmarking, where a new random sequence is drawn for each experimental trial. The advantages of full randomization include smaller confidence intervals on the inferred step error, the ability to use maximum likelihood analysis without heuristics, straightforward optimization of the sequence lengths, and the ability to model and measure behaviors that go beyond the typical assumption of time-independent error rates. We discuss models of time-dependent or non-Markovian errors that generalize the basic RB model of a single exponential decay of the success probability. For any of these models, we implement a concrete protocol to minimize the uncertainty of the estimated parameters given a fixed time constraint on the complete experiment, and we implement a maximum likelihood analysis. We consider several previously published experiments and determine the potential for improvements with optimized full randomization. We experimentally observe such improvements in Clifford randomized benchmarking experiments on a single trapped ion qubit at the National Institute of Standards and Technology (NIST). For an experiment with uniform lengths and intentionally repeated sequences the step error was $2.42^{+0.30}_{-0.22}\times 10^{-5}$, and for an optimized fully randomized experiment of the same total duration the step error was $2.57^{+0.07}_{-0.06}\times 10^{-5}$. We find a substantial decrease in the uncertainty of the step error as a result of optimized fully randomized benchmarking.

Device-independent random number generators can be certified unhackable simply by the impossibility of superluminal signaling. We demonstrate a novel protocol to transparently operationalize this security through a public randomness beacon service.

We develop a method to estimate the sum of conditional means of a sequence of random variables given our access to only a subsequence by spot-checking. The method works with non-independent-and-identically-distributed (non-i.i.d.) random variables and can be applied for certifying ongoing quantum information tasks.

Joint Gaussian measurements of two quantum systems are important for quantum communication between remote parties and are often used in continuous-variable teleportation or entanglement-swapping protocols. Many of the errors in real-world implementations can be modeled by independent Gaussian error channels acting prior to measurement. In this work we study independent single-mode Gaussian error channels on two modes A and B that take place prior to a joint Gaussian measurement. We determine the set of pairs of such channels that render all Gaussian measurements separable, and therefore unsuitable for entanglement swapping or teleportation of arbitrary input states. For example, if the error channels are loss with parameters lA,lB followed by added noise with parameters nA,nB then all Gaussian measurements are separable if and only if lA+lB+nA+nB≥1.

For many quantum systems intended for information processing, one detects the logical state of a qubit by integrating a continuously observed quantity over time. For example, ion and atom qubits are typically measured by driving a cycling transition and counting the number of photons observed from the resulting fluorescence. Instead of recording only the total observed count in a fixed time interval, one can observe the photon arrival times and get a state detection advantage by using the temporal structure in a model such as a Hidden Markov Model. We study what further advantage may be achieved by applying pulses to adaptively transform the state during the observation. We give a three-state example where adaptively chosen transformations yield a clear advantage, and we compare performances on an ion example, where we see improvements in some regimes. We provide a software package that can be used for exploration of temporally resolved strategies with and without adaptively chosen transformations.

We propose and demonstrate a protocol for high-fidelity indirect readout of trapped ion hyperfine qubits, where the state of a ^{9}Be^{+} qubit ion is mapped to a ^{25}Mg^{+} readout ion using laser-driven Raman transitions. By partitioning the ^{9}Be^{+} ground-state hyperfine manifold into two subspaces representing the two qubit states and choosing appropriate laser parameters, the protocol can be made robust to spontaneous photon scattering errors on the Raman transitions, enabling repetition for increased readout fidelity. We demonstrate combined readout and back-action errors for the two subspaces of 1.2_{-0.6}^{+1.1}×10^{-4} and 0_{-0}^{+1.9}×10^{-5} with 68% confidence while avoiding decoherence of spectator qubits due to stray resonant light that is inherent to direct fluorescence detection.

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. We implement a new scheme for universal control of trapped ion qubits based on microwave magnetic fields and rf magnetic field gradients instead of laser beams, and create entangled states whose fidelity is statistically indistinguishable from the highest-fidelity laser-beam-generated entangled states reported to date. We describe the rf and microwave electronics used, and discuss avenues for using this scheme in a large-scale trapped ion quantum processor.