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

TL;DR

This paper introduces a method to optimize entanglement verification in quantum states using limited samples, balancing confidence and efficiency with finite data analysis.

Contribution

It presents a novel analytical approach combining Frequentist and Bayesian methods to improve entanglement detection accuracy with small datasets.

Findings

Optimized entanglement verification with up to 20 copies.

Balances validity and efficiency in finite-sample scenarios.

Applicable to arbitrary entanglement witnesses.

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.