# Proof of "Entropy of preparation" in Von Neumann entropy

1. Jan 23, 2015

### 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{|φx〉,px}, so that the density matrix is ρ = ∑pxx〉<φx|
Then, H(X)≥S(ρ)
where H stands for Shannon entropy of probability {px} and S stands for Von Neumann entropy.

2. Jan 23, 2015

### 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.