What is the definition of log(P) in the von neumann entropy formula?

In summary, the von neumann entropy is defined as S(P) = -tr(P*log(P)), where log(P) is defined as -\sum_{n=1}^\infty \frac{1}{n}(I-A)^n and S(P) = H({k}) when P is diagonalized with eigenvalues {k}. The definition is similar to Gibbs' entropy and is expressed as S:=-k\langle \ln\hat{\rho}\rangle_{\hat{\rho}}.
  • #1
trosten
47
0
Let P be a density matrix. Then the von neumann entropy is defined as
S(P) = -tr(P*log(P))

But how is log(P) defined ?

--edit--
found the answer. the trace is independent of representation so that if P is diagonalized with eigenvalues {k} then S(P) = H({k}) where H is the shannon entropy.
 
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  • #2
[tex]ln(P) = -\sum_{n=1}^\infty \frac{1}{n}(I-A)^n[/tex]
 
  • #3
trosten said:
found the answer. the trace is independent of representation so that if P is diagonalized with eigenvalues {k} then S(P) = H({k}) where H is the shannon entropy.
right. Usually easier by the way !
 
  • #4
The definition is really

[tex] S:=-k\langle \ln\hat{\rho}\rangle_{\hat{\rho}} [/tex]

,quite similar to Gibbs' entropy.

Daniel.
 

1. What is the definition of Von Neumann entropy?

Von Neumann entropy, also known as Shannon entropy or simply entropy, is a measure of the uncertainty or randomness in a system. It is commonly used in information theory and statistical mechanics to measure the amount of information contained in a system.

2. How is Von Neumann entropy calculated?

Von Neumann entropy is calculated using the formula S = -Tr(ρln(ρ)), where S is the entropy, ρ is the density matrix of the system, and ln is the natural logarithm. This formula is derived from the von Neumann entropy formula for a discrete probability distribution.

3. What is the significance of Von Neumann entropy in quantum mechanics?

In quantum mechanics, Von Neumann entropy is used to measure the degree of entanglement between two or more particles. It is also used to quantify the information contained in a quantum state, which is important for understanding the behavior of quantum systems.

4. How does Von Neumann entropy relate to the second law of thermodynamics?

The second law of thermodynamics states that the total entropy of a closed system never decreases over time. Von Neumann entropy is a measure of the randomness or disorder in a system, and thus it is closely related to the second law of thermodynamics.

5. Can Von Neumann entropy be negative?

Yes, Von Neumann entropy can be negative. This occurs when the density matrix of the system has negative eigenvalues. Negative entropy is typically associated with systems that are highly ordered and predictable.

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