Entanglement Certification in Witness Experiments with Arbitrary Noise

Entanglement is a fundamental property of quantum mechanical systems and an essential resource for quantum network applications. Carefully certifying entanglement is therefore an important task that is often achieved using entanglement witness experiments. Given only finite trials, the statistical confidence 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 derive a method to certify entanglement without putting any assumptions on the state at each trial. This allows for arbitrarily correlated noise. We quantify the confidence using the p-value, the probability of obtaining data as extreme as the observed experimental data with only separable states. We give a formal abstract model of witness experiments that takes into account imperfect measurement devices and random number generators. We provide a simple method for sound data collection and show how to compute an upper bound to the p-value from the observed measurement data and device characteristics. We illustrate the use of our method with detailed examples based on realistic simulation of NV centres. This method applies to any witness experiment in general.