Von Neumann Entropy of GHZ state

In summary, the conversation discusses the simplest form of a Greenberger-Horne-Zeilinger (GHZ) state and its density matrix, as well as the reduced density matrix of a qubit. The eigenvalue equation and Von Neumann entropy of the reduced density matrix are also mentioned. It is noted that the off-diagonal terms in the density matrix cancel when finding the reduced matrix through tracing.
  • #1
neu
230
3
I just wanted to run this working by some of you.

Simplest Greenberger-Horne-Zeilinger state (entagled) state is:

[tex]\mid GHZ \rangle = \frac{1}{\sqrt{2}}\left(\mid 0 \rangle_{A}\mid 0 \rangle_{B}\mid 0 \rangle_{C}+\mid 1 \rangle_{A}\mid 1 \rangle_{B}\mid 1 \rangle_{C}\right)[/tex]

density matrix is:
[tex] \rho = \frac{1}{2} \left( \mid 0 \rangle \langle 0 \mid_{A}\mid 0 \rangle \langle 0 \mid_{B}\mid 0 \rangle \langle 0 \mid_{C} + \mid 1 \rangle \langle 1 \mid_{A}\mid 1 \rangle \langle 1 \mid_{B}\mid 1 \rangle \langle 1 \mid_{C} \right) [/tex]

reduced density matrix of qubit A:

[tex] \rho_{A} = Tr_{B}\left(Tr_{C}\rho\right) = \frac{1}{2} \left( \mid 0 \rangle \langle 0 \mid_{A}Tr\left(\mid 0 \rangle \langle 0 \mid_{B}\right)Tr\left(\mid 0 \rangle \langle 0 \mid_{C}\right) + \mid 1 \rangle \langle 1 \mid_{A}Tr\left(\mid 1 \rangle \langle 1 \mid_{B}\right)Tr\left(\mid 1 \rangle \langle 1 \mid_{C}\right) \right) [/tex]

[tex] \rho_{A} = \frac{1}{2}\left( \mid 0 \rangle \langle 0 \mid_{A} + \mid 1 \rangle \langle 1 \mid_{A}\right) = \frac{1}{2}
\left[\left(
\begin{array}{ c c }
1 & 0 \\
0 & 0
\end{array}\right) +
\left(
\begin{array}{ c c }
0 & 0\\
0 & 1
\end{array}\right)\right]
[/tex]

So the eigenvalue equation of [tex]\rho_{A}[/tex] is :
[tex]
\mid
\begin{array}{ c c }
\frac{1}{2}-\lambda & 0\\
0 & \frac{1}{2}-\lambda
\end{array}\mid = 0
[/tex]

so [tex]\lambda = \frac{1}{2}[/tex] and Von neumann entropy [tex] S(\rho_{A}) = - \Sigma_{i} \lambda_{i} log_{2} \lambda_{i} [/tex] is:

[tex] 2^{-2S(\rho_{A})} = \frac{1}{2} [/tex]

So [tex] S(\rho_{A}) = \frac{1}{2}[/tex]
 
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  • #2
oui ou non?
 
  • #3
No. The density matrix has off-diagonal terms as well.
 
  • #4
genneth said:
No. The density matrix has off-diagonal terms as well.

Yeah I realize this, but they cancel when finding the reduced matrix from the tracing.

So get same result.

Thanks I get it anyway now; I've gone over it a few times
 

1. What is the Von Neumann Entropy of GHZ state?

The Von Neumann Entropy of GHZ state is a measure of the amount of uncertainty or randomness in a quantum system known as the GHZ state. It is named after physicist John von Neumann and is commonly used in quantum information theory to quantify the amount of entanglement present in a quantum state.

2. How is the Von Neumann Entropy of GHZ state calculated?

The Von Neumann Entropy of GHZ state is calculated by taking the trace of the density matrix of the state and applying the Von Neumann entropy formula, -Tr(ρlog(ρ)). The density matrix is a mathematical representation of the quantum state and the trace operator sums up the diagonal elements of the matrix.

3. What does a high or low Von Neumann Entropy of GHZ state indicate?

A high Von Neumann Entropy of GHZ state indicates a high degree of entanglement in the quantum system. This means that the subsystems in the state are strongly correlated and their measurements are highly correlated as well. On the other hand, a low Von Neumann Entropy of GHZ state indicates a low degree of entanglement and the subsystems are not strongly correlated.

4. How does the Von Neumann Entropy of GHZ state relate to other measures of entanglement?

The Von Neumann Entropy of GHZ state is a type of entanglement entropy, which is a class of measures used to quantify the amount of entanglement in a quantum state. It is closely related to other entanglement measures such as the Renyi entropy and the Tsallis entropy, and can be used to calculate these quantities in certain cases.

5. What are the practical applications of the Von Neumann Entropy of GHZ state?

The Von Neumann Entropy of GHZ state has various applications in quantum information and quantum computing. It is used to quantify the amount of entanglement in a quantum system, which is a crucial resource for many quantum information processing tasks such as quantum teleportation, quantum cryptography, and quantum error correction. It is also used in the study of quantum phase transitions and quantum criticality in many-body systems.

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