On the definition of Von Neumann entropy

[univector]

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
S (\rho) = - tr(\rho \log \rho)
where \rho 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.

 

[Fredrik]

Note that

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

Integrate.

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

Now set y=1+x.

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

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

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

for all matrices A such that the series converges. More information here.

 

[univector]

Hi Fredrik,

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

Is this the way people calculate S(\rho)?

 

[Avodyne]

Yes.