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Witnessing Entanglement in Experiments with Arbitrary Noise

Bas Dirkse, Matteo Pompili, Ronald Hanson, Michael Walter, Stephanie Wehner

Published Feb 27, 2020• 0 citations
OpenAlex

Abstract

The purpose of an entanglement witness experiment is to certify the creation of an entangled state from a finite number of trials. The statistical confidence of such an experiment is typically expressed as the number of observed standard deviations of witness violations. This method implicitly assumes that the noise is well-behaved so that the central limit theorem applies. In this work, we propose two methods to analyze witness experiments where the states can be subject to arbitrarily correlated noise. Our first method is a rejection experiment, in which we certify the creation of entanglement by rejecting the hypothesis that the experiment can only produce separable states. We quantify the statistical confidence by a p-value, which can be interpreted as the likelihood that the observed data is consistent with the hypothesis that only separable states can be produced. Hence a small p-value implies large confidence in the witnessed entanglement. The method applies to general witness experiments and can also be used to witness genuine multipartite entanglement. Our second method is an estimation experiment, in which we estimate and construct confidence intervals for the average witness value. This confidence interval is statistically rigorous in the presence of correlated noise. The method applies to general estimation problems, including fidelity estimation. To account for systematic measurement and random setting generation errors, our model takes into account device imperfections and we show how this affects both methods of statistical analysis. Finally, we illustrate the use of our methods with detailed examples based on a simulation of NV centers.

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The purpose of an entanglement witness experiment is to certify the creation of an entangled state from a finite number of trials. The statistical confidence of such an experiment is typically expressed as the number of observed standard deviations of witness violations. This method implicitly assumes that the noise is well-behaved so that the central limit theorem applies. In this work, we propose two methods to analyze witness experiments where the states can be subject to arbitrarily correlated noise. Our first method is a rejection experiment, in which we certify the creation of entanglement by rejecting the hypothesis that the experiment can only produce separable states. We quantify the statistical confidence by a p-value, which can be interpreted as the likelihood that the observed data is consistent with the hypothesis that only separable states can be produced. Hence a small p-value implies large confidence in the witnessed entanglement. The method applies to general witness experiments and can also be used to witness genuine multipartite entanglement. Our second method is an estimation experiment, in which we estimate and construct confidence intervals for the average witness value. This confidence interval is statistically rigorous in the presence of correlated noise. The method applies to general estimation problems, including fidelity estimation. To account for systematic measurement and random setting generation errors, our model takes into account device imperfections and we show how this affects both methods of statistical analysis. Finally, we illustrate the use of our methods with detailed examples based on a simulation of NV centers.
Abstract The purpose of an entanglement witness experiment is to certify the creation of an entangled state from a finite number of trials. The statistical confidence of such an experiment is typically expressed as the number of observed standard deviations of witness violations. This method implicitly assumes that the noise is well-behaved so that the central limit theorem applies. In this work, we propose two methods to analyze witness experiments where the states can be subject to arbitrarily correlated noise. Our first method is a rejection experiment, in which we certify the creation of entanglement by rejecting the hypothesis that the experiment can only produce separable states. We quantify the statistical confidence by a p-value, which can be interpreted as the likelihood that the observed data is consistent with the hypothesis that only separable states can be produced. Hence a small p-value implies large confidence in the witnessed entanglement. The method applies to general witness experiments and can also be used to witness genuine multipartite entanglement. Our second method is an estimation experiment, in which we estimate and construct confidence intervals for the average witness value. This confidence interval is statistically rigorous in the presence of correlated noise. The method applies to general estimation problems, including fidelity estimation. To account for systematic measurement and random setting generation errors, our model takes into account device imperfections and we show how this affects both methods of statistical analysis. Finally, we illustrate the use of our methods with detailed examples based on a simulation of NV centers.
Standard procedures for entanglement detection assume that experimenters can exactly implement specific quantum measurements. Here, we depart from such idealizations and investigate, in both theory and experiment, the detection of genuine multipartite entanglement when measurements are subject to small imperfections. For arbitrary qubits number n, we construct multipartite entanglement witnesses where the detrimental influence of the imperfection is independent of n. In a tabletop four-partite photonic experiment, we demonstrate first how a small amount of alignment error can undermine the conclusions drawn from standard entanglement witnesses and then perform the correction analysis. Furthermore, since we consider quantum devices that are trusted but not perfectly controlled, we showcase advantages in terms of noise resilience as compared to device-independent models.
Abstract Detecting entanglement in multipartite quantum states is an inherently probabilistic process, typically with a few measured samples. The level of confidence in entanglement detection quantifies the scheme’s validity via the probability that the signal comes from a separable state, offering a meaningful figure of merit for big datasets. Yet, with limited samples, avoiding experimental data misinterpretations requires considering not only the probabilities concerning separable states but also the probability that the signal came from an entangled state, i.e. the detection scheme’s efficiency. We demonstrate this explicitly and apply a general method to optimize both the validity and the efficiency in small data sets providing examples using at most 20 state copies. The method is based on an analytical model of finite statistics effects on correlation functions which takes into account both a Frequentist as well as a Bayesian approach and is applicable to arbitrary entanglement witnesses.
Quantum mechanics predicts the existence of correlations between composite systems that, although puzzling to our physical intuition, enable technologies not accessible in a classical world. Notwithstanding, there is still no efficient general method to theoretically quantify and experimentally detect entanglement of many qubits. Here we propose to detect entanglement by measuring the statistical response of a quantum system to an arbitrary nonlocal parametric evolution. We witness entanglement without relying on the tomographic reconstruction of the quantum state, or the realization of witness operators. The protocol requires two collective settings for any number of parties and is robust against noise and decoherence occurring after the implementation of the parametric transformation. To illustrate its user friendliness we demonstrate multipartite entanglement in different experiments with ions and photons by analyzing published data on fidelity visibilities and variances of collective observables.
Any experiment attempting to verify the presence of entanglement in a physical system can only generate a finite amount of data. The statement that entanglement was present in the system can thus never be issued with certainty, requiring instead a statistical analysis of the data. Because entanglement plays a central role in the performance of quantum devices, it is crucial to make statistical claims in entanglement verification experiments that are reliable and have a clear interpretation. In this work, we apply recent results by Christandl and Renner [Phys. Rev. Lett. 109, 120403 (2012)] to construct a reliable entanglement verification procedure based on the concept of confidence regions. The statements made do not require the specification of a prior distribution, the assumption of independent measurements, or the assumption of an independent and identically distributed source of states. Moreover, we develop numerical tools that are necessary to employ this approach in practice, rendering the procedure ready to be applied to current experiments. We demonstrate this technique by analyzing the data of a photonic experiment generating two-photon states whose entanglement is verified with the use of an accessible nonlinear witness.
Suppose an experimentalist wishes to verify that his apparatus produces entangled quantum states. A finite amount of data cannot conclusively demonstrate entanglement, so drawing conclusions from real-world data requires statistical reasoning. We propose a reliable method to quantify the weight of evidence for (or against) entanglement, based on a likelihood ratio test. Our method is universal in that it can be applied to any sort of measurements. We demonstrate the method by applying it to two simulated experiments on two qubits. The first measures a single entanglement witness, while the second performs a tomographically complete measurement.