Let’s go back to our spin-1/2 friends and construct the density matrix for a collection of spin-1/2 particles. Let’s say that we have a partially polarized beam for which a fraction of 3/4 are polarized in the +z direction and 1/4 of the sample are polarized along the +x direction. Our density matrix of this system is

ρ =34|↑z⟩⟨↑z |+14|↑x⟩⟨↑x | . (12.56)

It’s easy enough to calculate the von Neumann entropy of this density matrix, as it is already in diagonal form. We find SvN = −tr(ρ logρ) = −34log34− 14log14≃ 0.562. (12.57)