Proof of "Entropy of preparation" in Von Neumann entropy

[nochemala]

How should I prove this?

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.

 

[jfizzix]

If you can find it, on page 519 in Nielsen and Chuang's (Quantum Computation and quantum information, 10th anniversary edition), they show how to do it.

In short, you express |\phi_{x}\rangle in terms of the eigenstates of the density matrix \rho. Since the entropy of a mixture of states is larger than the corresponding mixture of entropies, you can eventually find:
S(\rho)\leq H(X)+\sum_{x}p_{x}S(\rho_{x})
Since the states in the ensemble are pure, this would give you your result.