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On the definition of Von Neumann entropy

  1. Jul 12, 2009 #1
    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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  3. Jul 12, 2009 #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.
     
    Last edited: Jul 12, 2009
  4. Jul 13, 2009 #3
    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]?
     
  5. Jul 13, 2009 #4

    Avodyne

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