I know how to get Von Neumann entropy from a density matrix. I want to get a real number from measurements that give real numbers as outcomes. (there are complex numbers in a density matrix). So suppose Charlie sends 1000 pairs of particles in the same state to Bob and Alice. They agree to measure the spin along two directions differing by an angle theta. (They can agree with Charlie for another set of 1000 pairs with another angle theta' if they want). When they meet again, they compare their results and build a square table with their numbers of uu ud du and dd (the same for theta') How can they estimate the Von Neuman entropy and know if Charlie sent them strongly entangled pairs?