Proof of "Entropy of preparation" in Von Neumann entropy

In summary, to prove that H(X)≥S(ρ), you first need to express the state |\phi_{x}\rangle in terms of the eigenstates of the density matrix \rho. Then, using the fact that the entropy of a mixture of states is larger than the corresponding mixture of entropies, you can eventually find that S(\rho)\leq H(X)+\sum_{x}p_{x}S(\rho_{x}). This will give you the desired result since the states in the ensemble are pure.
  • #1
nochemala
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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 = ∑px|φx〉<φx|
Then, H(X)≥S(ρ)
where H stands for Shannon entropy of probability {px} and S stands for Von Neumann entropy.
 
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  • #2
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 [itex]|\phi_{x}\rangle[/itex] in terms of the eigenstates of the density matrix [itex]\rho[/itex]. Since the entropy of a mixture of states is larger than the corresponding mixture of entropies, you can eventually find:
[itex]S(\rho)\leq H(X)+\sum_{x}p_{x}S(\rho_{x})[/itex]
Since the states in the ensemble are pure, this would give you your result.
 
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FAQ: Proof of "Entropy of preparation" in Von Neumann entropy

1. What is the concept of "Entropy of preparation" in Von Neumann entropy?

The entropy of preparation in Von Neumann entropy is a measure of the amount of uncertainty or randomness in the state of a quantum system. It is based on the idea that the more uncertain we are about the state of a system, the higher its entropy and the less information we have about it.

2. How is the "Entropy of preparation" calculated in Von Neumann entropy?

The entropy of preparation is calculated using the Von Neumann entropy formula, which takes into account the density matrix of a quantum system. The density matrix is a mathematical representation of the state of a quantum system, and the Von Neumann entropy formula uses its eigenvalues to calculate the entropy of preparation.

3. What is the significance of "Entropy of preparation" in quantum information theory?

The "Entropy of preparation" is a key concept in quantum information theory, as it helps us understand the uncertainty and randomness in quantum systems. It is also used in various quantum computing algorithms, such as quantum error correction, to minimize the impact of noise and uncertainty in quantum systems.

4. How does "Entropy of preparation" relate to the concept of entanglement in quantum mechanics?

The "Entropy of preparation" is closely related to the concept of entanglement in quantum mechanics. Entanglement is a phenomenon where two or more quantum systems become connected in such a way that their states are dependent on each other. As the level of entanglement increases, the "Entropy of preparation" also increases, indicating a higher level of uncertainty in the quantum system.

5. Can the "Entropy of preparation" be used to measure the complexity of a quantum system?

Yes, the "Entropy of preparation" can be used to measure the complexity of a quantum system. As the entropy increases, it indicates a higher level of uncertainty and randomness in the system, which can be seen as a measure of its complexity. This concept has been used in various fields such as quantum computing and quantum information theory to study and analyze complex quantum systems.

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