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Berthold-Georg Englert

Publications and source records attributed to Berthold-Georg Englert.

At least 37 records · Page 2Linked to original sources
arXiv2019

Proper error bars for self-calibrating quantum tomography

Self-calibrating quantum state tomography aims at reconstructing the unknown quantum state and certain properties of the measurement devices from the same data. Since the estimates of the state and device parameters come from the same data, one should employ a joint estimation scheme, including the construction and reporting of joint state-device error regions to quantify uncertainty. We explain how to do this naturally within the framework of optimal error regions. As an illustrative example, we apply our procedure to the double-crosshair measurement of the BB84 scenario in quantum cryptography and so reconstruct the state and estimate the detection efficiencies simultaneously and reliably. We also discuss the practical situation of a satellite-based quantum key distribution scheme, for which self-calibration and proper treatment of the data are necessities.

quant-ph↗
arXiv2019

Very strong evidence in favor of quantum mechanics and against local hidden variables from a Bayesian analysis

The data of four recent experiments --- conducted in Delft, Vienna, Boulder, and Munich with the aim of refuting nonquantum hidden-variables alternatives to the quantum-mechanical description --- are evaluated from a Bayesian perspective of what constitutes evidence in statistical data. We find that each of the experiments provides strong, or very strong, evidence in favor of quantum mechanics and against the nonquantum alternatives. This Bayesian analysis supplements the previous non-Bayesian ones, which refuted the alternatives on the basis of small p-values, but could not support quantum mechanics.

quant-ph↗
arXiv2018

Experimental investigation of high-dimensional quantum key distribution protocols with twisted photons

Quantum key distribution is on the verge of real world applications, where perfectly secure information can be distributed among multiple parties. Several quantum cryptographic protocols have been theoretically proposed and independently realized in different experimental conditions. Here, we develop an experimental platform based on high-dimensional orbital angular momentum states of single photons that enables implementation of multiple quantum key distribution protocols with a single experimental apparatus. Our versatile approach allows us to experimentally survey different classes of quantum key distribution techniques, such as the 1984 Bennett \& Brassard (BB84), tomographic protocols including the six-state and the Singapore protocol, and to investigate, for the first time, a recently introduced differential phase shift (Chau15) protocol using twisted photons. This enables us to experimentally compare the performance of these techniques and discuss their benefits and deficiencies in terms of noise tolerance in different dimensions.

quant-ph↗
arXiv2018

The Statistical Atom

This review is the updated and enlarged version of a talk delivered by J. S. on the occasion of the 1982 meeting of Nobel laureates at Lindau, and of talks given by B.-G. E. at several West German universities and Max Planck institutes in 1984.

quant-ph↗
arXiv2018

Checking the Model and the Prior for the Constrained Multinomial

The multinomial model is one of the simplest statistical models. When constraints are placed on the possible values for the probabilities, however, it becomes much more difficult to deal with. Model checking and checking for prior-data conflict is considered here for such models. A theorem is proved that establishes the consistency of the check on the prior. Applications are presented to models that arise in quantum state estimation as well as the Bayesian analysis of models for ordered probabilities.

math.ST↗
arXiv2018

Systematic corrections to the Thomas-Fermi approximation without a gradient expansion

We improve on the Thomas-Fermi approximation for the single-particle density of fermions by introducing inhomogeneity corrections. Rather than invoking a gradient expansion, we relate the density to the unitary evolution operator for the given effective potential energy and approximate this operator by a Suzuki-Trotter factorization. This yields a hierarchy of approximations, one for each approximate factorization. For the purpose of a first benchmarking, we examine the approximate densities for a few cases with known exact densities and observe a very satisfactory, and encouraging, performance. As a bonus, we also obtain a simple fourth-order leapfrog algorithm for the symplectic integration of classical equations of motion.

cond-mat.quant-gas↗
arXiv2017

Unambiguous path discrimination in a two-path interferometer

When a photon is detected after passing through an interferometer one might wonder which path it took, and a meaningful answer can only be given if one has the means of monitoring the photon's whereabouts. We report the realization of a single-photon experiment for a two-path interferometer with path marking. In this experiment, the path of a photon ("signal") through a Mach--Zehnder interferometer becomes known by unambiguous discrimination between the two paths. We encode the signal path in the polarization state of a partner photon ("idler") whose polarization is examined by a three-outcome measurement: one outcome each for the two signal paths plus an inconclusive outcome. Our results agree fully with the theoretical predictions from a common-sense analysis of what can be said about the past of a quantum particle: The signals for which we get the inconclusive result have full interference strength, as their paths through the interferometer \emph{cannot be known}; and every photon that emerges from the dark output port of the balanced interferometer has a \emph{known path}.

quant-ph↗
arXiv2017

Past of a quantum particle: Common sense prevails

We analyze Vaidman's three-path interferometer with weak path marking [Phys. Rev. A 87, 052104 (2013)] and find that common sense yields correct statements about the particle's path through the interferometer. This disagrees with the original claim that the particles have discontinuous trajectories at odds with common sense. In our analysis, "the particle's path" has operational meaning as acquired by a path-discriminating measurement. For a quantum-mechanical experimental demonstration of the case, one should perform a single-photon version of the experiment by Danan et al. [Phys. Rev. Lett. 111, 240402 (2013)] with unambiguous path discrimination. We present a detailed proposal for such an experiment.

quant-ph↗
arXiv2017

Airy-averaged gradient corrections for two-dimensional fermion gases

Building on the discussion in PRA 93, 042510 (2016), we present a systematic derivation of gradient corrections to the kinetic-energy functional and the one-particle density, in particular for two-dimensional systems. We derive the leading gradient corrections from a semiclassical expansion based on Wigner's phase space formalism and demonstrate that the semiclassical kinetic-energy density functional at zero temperature cannot be evaluated unambiguously. In contrast, a density-potential functional description that effectively incorporates interactions provides unambiguous gradient corrections. Employing an averaging procedure that involves Airy functions, thereby partially resumming higher-order gradient corrections, we facilitate a smooth transition of the particle density into the classically forbidden region of arbitrary smooth potentials. We find excellent agreement of the semiclassical Airy-averaged particle densities with the exact densities for very low but finite temperatures, illustrated for a Fermi gas with harmonic potential energy. We furthermore provide criteria for the applicability of the semiclassical expansions at low temperatures. Finally, we derive a well-behaved ground-state kinetic-energy functional, which improves on the Thomas-Fermi approximation.

cond-mat.quant-gas↗
arXiv2016

Random samples of quantum states: Online resources

This is the documentation for generating random samples from the quantum state space in accordance with a specified distribution, associated with this webpage: http://tinyurl.com/QSampling . Ready-made samples (each with at least a million points) from various distributions are available for download, or one can generate one's own samples from a chosen distribution using the provided source codes. The sampling relies on the Hamiltonian Monte Carlo algorithm as described in New J. Phys. 17, 043018 (2015). The random samples are reposited in the hope that they would be useful for a variety of tasks in quantum information and quantum computation. Constructing credible regions for tomographic data, optimizing a function over the quantum state space with a complicated landscape, testing the typicality of entanglement among states from a multipartite quantum system, or computing the average of some quantity of interest over a subset of quantum states are but some exemplary applications among many.

quant-ph↗
arXiv2016

Optimal error intervals for properties of the quantum state

Quantum state estimation aims at determining the quantum state from observed data. Estimating the full state can require considerable efforts, but one is often only interested in a few properties of the state, such as the fidelity with a target state, or the degree of correlation for a specified bipartite structure. Rather than first estimating the state, one can, and should, estimate those quantities of interest directly from the data. We propose the use of optimal error intervals as a meaningful way of stating the accuracy of the estimated property values. Optimal error intervals are analogs of the optimal error regions for state estimation [New J. Phys. 15, 123026 (2013)]. They are optimal in two ways: They have the largest likelihood for the observed data and the pre-chosen size, and are the smallest for the pre-chosen probability of containing the true value. As in the state situation, such optimal error intervals admit a simple description in terms of the marginal likelihood for the data for the properties of interest. Here, we present the concept and construction of optimal error intervals, report on an iterative algorithm for reliable computation of the marginal likelihood (a quantity difficult to calculate reliably), explain how plausible intervals --- a notion of evidence provided by the data --- are related to our optimal error intervals, and illustrate our methods with single-qubit and two-qubit examples.

quant-ph↗
arXiv2016

Checking for prior-data conflict using prior to posterior divergences

When using complex Bayesian models to combine information, the checking for consistency of the information being combined is good statistical practice. Here a new method is developed for detecting prior-data conflicts in Bayesian models based on comparing the observed value of a prior to posterior divergence to its distribution under the prior predictive distribution for the data. The divergence measure used in our model check is a measure of how much beliefs have changed from prior to posterior, and can be thought of as a measure of the overall size of a relative belief function. It is shown that the proposed method is intuitive, has desirable properties, can be extended to hierarchical settings, and is related asymptotically to Jeffreys' and reference prior distributions. In the case where calculations are difficult, the use of variational approximations as a way of relieving the computational burden is suggested. The methods are compared in a number of examples with an alternative but closely related approach in the literature based on the prior predictive distribution of a minimal sufficient statistic.

stat.ME↗
arXiv2016

Leading gradient correction to the kinetic energy for two-dimensional fermion gases

Density functional theory (DFT) is notorious for the absence of gradient corrections to the two-dimensional (2D) Thomas-Fermi kinetic-energy functional; it is widely accepted that the 2D analog of the 3D von Weizsäcker correction vanishes, together with all higher-order corrections. Contrary to this long-held belief, we show that the leading correction to the kinetic energy does not vanish, is unambiguous, and contributes perturbatively to the total energy. This insight emerges naturally in a simple extension of standard DFT, which has the effective potential energy as a functional variable on equal footing with the single-particle density.

cond-mat.quant-gas↗
arXiv2016

Implementing a neutral-atom controlled-phase gate with a single Rydberg pulse

One can implement fast two-qubit entangling gates by exploiting the Rydberg blockade. Although various theoretical schemes have been proposed, experimenters have not yet been able to demonstrate two-atom gates of high fidelity due to experimental constraints. We propose a novel scheme, which only uses a single Rydberg pulse illuminating both atoms, for the construction of neutral-atom controlled-phase gates. In contrast to the existing schemes, our approach is simpler to implement and requires neither individual addressing of atoms nor adiabatic procedures. With parameters estimated based on actual experimental scenarios, a gate fidelity higher than 0.99 is achievable.

quant-ph↗
arXiv2015

Measuring Quantum Correlations using Lossy Photon-Number-Resolving Detectors with Saturation

The variance of difference of photocounts (VDPs) is an established measure of quantum correlations for quantum states of light. It enables us to discriminate between the classical correlation of a two-mode coherent state and the quantum correlation of a twin-beam state. We study the effect of loss and saturation of the photon-number-resolving detector on the measurement of the VDPs. An analytic function is derived for this variance, both for the coherent and the twin-beam states. It is found that the VDPs is no longer a reliable entanglement measure in the nonlinear regime of the detector response but it remains useful in some range of values of average photon numbers of the incident light. We also quantify the linear regime of the detector with saturation which will be useful for calibration of the detector quantum efficiency.

quant-ph↗
arXiv2015

Optimality of entropic uncertainty relations

The entropic uncertainty relation proven by Maassen and Uffink for arbitrary pairs of two observables is known to be non-optimal. Here, we call an uncertainty relation optimal, if the lower bound can be attained for any value of either of the corresponding uncertainties. In this work we establish optimal uncertainty relations by characterising the optimal lower bound in scenarios similar to the Maassen-Uffink type. We disprove a conjecture by Englert et al. and generalise various previous results. However, we are still far from a complete understanding and, based on numerical investigation and analytical results in small dimension, we present a number of conjectures.

quant-ph↗
arXiv2015

Monte Carlo sampling from the quantum state space. I

High-quality random samples of quantum states are needed for a variety of tasks in quantum information and quantum computation. Searching the high-dimensional quantum state space for a global maximum of an objective function with many local maxima or evaluating an integral over a region in the quantum state space are but two exemplary applications of many. These tasks can only be performed reliably and efficiently with Monte Carlo methods, which involve good samplings of the parameter space in accordance with the relevant target distribution. We show how the standard strategies of rejection sampling, importance sampling, and Markov-chain sampling can be adapted to this context, where the samples must obey the constraints imposed by the positivity of the statistical operator. For a comparison of these sampling methods, we generate sample points in the probability space for two-qubit states probed with a tomographically incomplete measurement, and then use the sample for the calculation of the size and credibility of the recently-introduced optimal error regions [see New J. Phys. 15 (2013) 123026]. Another illustration is the computation of the fractional volume of separable two-qubit states.

quant-ph↗