The Schmidt number of a quantum state cannot always be device-independently certified

Abstract: One of the great challenges of quantum foundations and quantum information theory is the characterisation of the relationship between entanglement and the violation of Bell inequalities. It is well known that in specific scenarios these two can behave differently, from local hidden-variable models for entangled quantum states in restricted Bell scenarios, to maximal violations of Bell inequalities not concurring with maximal entanglement. In this paper we put forward a simple proof that there exist quantum states, whose entanglement content, as measured by the Schmidt number, cannot be device-independently certified for all possible sequential measurements on any number of copies. While the bigger question: \textit{can the presence of entanglement always be device-independently certified?} remains open, we provide proof that quantifying entanglement device-independently is not always possible, even beyond the standard Bell scenario.