Zero-correlation entanglement vs. Schmidt rank

Abstract

In this paper, we find the minimal number of correlation tests required to certify independence of two classical random variables of arbitrary finite support. Moreover, we completely characterize the constraints implied by zero correlations upon an arbitrary heterogeneous quantum state, and we solve the conjecture proposed by Ohira in 2020. Finally, we find the first ever algorithm for computing (or checking) the Schmidt rank of any unknown pure quantum state using only zero-correlation tests, finding a sufficient amount of tests certifying separability or full entanglement.