High validity entanglement verification with finite copies of a quantum state

Detecting entanglement of multipartite quantum states is an inherently probabilistic process, typically with a small number of measured samples. The level of confidence in entanglement detection can be used to quantify the validity of the detection scheme via the probability that the measured signal is coming from a separable state, and provides a meaningful figure of merit for big data sets. Yet, for limited sample sizes, to avoid serious misinterpretations of the experimental results, one should not only consider the probability that a separable state gave rise to the measured signal but should also include information about the probability that the signal came from an entangled state, i.e. the efficiency of the detection scheme. We demonstrate this explicitly and propose a general method to optimize both the validity and the efficiency in small data sets. The method is applicable to arbitrary entanglement witnesses and is based on an analytical model of finite statistics effects on correlation functions. As an example, we derive the optimal number of measurement settings and distribution of clicks per setting that guarantee high validity and efficiency of entanglement verification with only 20 copies of a state. The analysis takes into account both a Frequentist as well as Bayesian approach.