On the definition of Von Neumann entropy

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I am confused by the definition of the Von Neumann entropy. In Nielson and Chung's book page 510, the Von Neumann entropy is defined as
[tex] S (\rho) = - tr(\rho \log \rho) [/tex]
where [tex] \rho [/tex] is the density matrix. What is the definition of the logrithm of a matrix? Is it some series expansion of a matrix, or an element-by-element logrithm?

Thanks.
 

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  • #2
Fredrik
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Note that

[tex]\frac{d}{dx}\log(1+x)=\frac{1}{1+x}=1-x+x^2-x^3+\cdots[/tex]

Integrate.

[tex]\log(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\cdots[/tex]

Now set y=1+x.

[tex]\log y=(y-1)-\frac{(y-1)^2}{2}+\cdots=\sum_{k=1}^\infty(-1)^{k+1}\frac{(y-1)^k}{k}[/tex]

This suggests that if A is a matrix, we can define log A by

[tex]\log A=\sum_{k=1}^\infty(-1)^{k+1}\frac{(A-I)^k}{k}[/tex]

for all matrices A such that the series converges. More information here.
 
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  • #3
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Hi Fredrik,

Thanks for the explanation. If the matrix [itex] \rho [/itex] is not diagonal, it is not trivial to calculate the matrix polynomial series. Instead, if we do a similary transform to diagonize [itex] \rho [/itex] first, things may become easier. Let the diagonal matrix be [itex] D [/itex]. Then by using the series expansion forward (for a matrix) and backward (for a number), we can come up with
[tex] S(\rho) = tr (D M) [/tex],
where
[tex] M = \left( \begin{array}{cccc} \log \lambda_1 & 0 & 0 & ... \\
0 & \log \lambda_2 & 0 & ... \\
... & ... & ... & ... \\
0 & ... & 0 & \log \lambda_n
\end{array} \right)[/tex]
where [itex] \lambda_i [/itex] is the ith engenvalue of matrix [itex] \rho [/itex] (or [itex] D [/itex]), and [itex] n [/itex] is the number of rows (also columns) of [itex] \rho [/itex] .

Is this the way people calculate [itex] S(\rho) [/itex]?
 
  • #4
Avodyne
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Yes.
 

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