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From John Preskill's quantum computation & quantum information lecture notes(chapter 5)

If a pure state is drawn randomly from the ensemble{|φ

_{x}〉,p

_{x}}, so that the density matrix is ρ = ∑p

_{x}|φ

_{x}〉<φ

_{x}|

Then, H(X)≥S(ρ)

where H stands for Shannon entropy of probability {p

_{x}} and S stands for Von Neumann entropy.