Let's say I have a pure state of the form: $$\psi = \sqrt{\frac{3}{9}} \lvert 0 \rangle + \sqrt{\frac{6}{9}} \lvert 1 \rangle$$

Then the density matrix representation would be: $$\rho = \psi \otimes \psi' = \begin{bmatrix}.3333&.4714\\4714&.6667\end{bmatrix}$$

Now, what would be the Von Neumann entropy of this matrix? I saw that the equation is:

$$S(\rho) = -\text{trace}(\rho \log \rho) = 1.3455.$$ But it's supposed to be 0, isn't it?

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    $\begingroup$ Did you make sure to do log as a matrix not element-wise? If you did it with a program, it often assumes you mean element-wise. $\endgroup$ – AHusain Mar 18 '19 at 2:45
  • $\begingroup$ Thanks AHusain, that was the issue I see. $\endgroup$ – Hasan Iqbal Mar 18 '19 at 3:46
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    $\begingroup$ @HasanIqbal If you've found your mistake, you could answer your own question too. :) $\endgroup$ – Sanchayan Dutta Mar 18 '19 at 18:53

It turns out to be a novice mistake. I was using matlab and this log is elementwise, as @Ahusain pointed out. We must take the matrix logarithm in Matlab which is denoted by $logm$. Then the calculation becomes: $$-\text{trace}(\rho \log m (\rho)) = \text{NaN}.$$ The reason is, we have to define $0 \times \log (0)$ as $0$ instead of $\text{NaN}$ which is the default behavior of Matlab. Another way to calculate it is the following:

As this density matrix is representing a pure state, it would have a diagonalization. I.e. it can be written as:

$$\rho = \sum_{j} \lambda_j \lvert \phi_j \rangle \langle \phi_j \lvert$$

Where, $\lambda_j$ are it's eigenvalues and $\lvert \phi_j \rangle$ are it's distinct eigenvectors. In which case, the von Neumann entropy is simply the shannon entropy of it's eigenvalues:

$$S(\rho) = H(\lambda_1, \lambda_2) = -0 \times \log(0) - 1 \times \log(1) = 0$$Using the above definition of $\log(0) = 0$.

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    $\begingroup$ You may use \times for the $\times$ multiplication symbol rather than $*$. Alternatively, use a $.$ dot. $\endgroup$ – Sanchayan Dutta Mar 19 '19 at 16:48

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