Valid and efficient entanglement verification with finite copies of a quantum state

Detecting entanglement in multipartite quantum states is an inherentlyprobabilistic process, typically with a few measured samples. The level ofconfidence in entanglement detection quantifies the scheme's validity via theprobability that the signal comes from a separable state, offering a meaningfulfigure of merit for big datasets. Yet, with limited samples, avoidingexperimental data misinterpretations requires considering not only theprobabilities concerning separable states but also the probability that thesignal came from an entangled state, i.e. the detection scheme's efficiency. Wedemonstrate this explicitly and apply a general method to optimize both thevalidity and the efficiency in small data sets providing examples using at most20 state copies. The method is based on an analytical model of finitestatistics effects on correlation functions which takes into account both aFrequentist as well as a Bayesian approach and is applicable to arbitraryentanglement witnesses.