Density matrix and von Neumann entropy - why does basis matter?

AI Thread Summary
The discussion centers on the confusion surrounding the calculation of von Neumann entropy for a pure state using different bases. The user initially computes the density matrix in the |0⟩, |1⟩ basis and incorrectly assumes the logarithm of the matrix can be applied elementwise, leading to an erroneous entropy value of 2. It is clarified that the logarithm of a matrix must be computed using proper matrix operations, not elementwise, which is valid only for diagonal matrices. The conclusion emphasizes that while the basis choice can affect the representation of the density matrix, the fundamental properties of the entropy remain consistent. Understanding the correct application of matrix logarithms resolves the confusion regarding the basis dependency in entropy calculations.
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Density matrix and von Neumann entropy -- why does basis matter?

I'm very confused by why I'm unable to correctly compute the von Neumann entropy

S = - \mathrm{Tr}(\rho \log_2{\rho})

for the pure state

| \psi \rangle = \left(|0\rangle + |1\rangle\right)/ \sqrt 2

Now, clearly the simplest thing to do is to express |\psi\rangle in the |+\rangle,|-\rangle basis, where it's clear that

\rho = |+\rangle\langle+|

In this basis, S = 0 as we'd expect for a pure state.

What I can't fathom (and I'm sure I'm missing something really obvious) is why if I evaluate the entropy in the |0\rangle,|1\rangle basis,

\rho = \begin{pmatrix}<br /> 1/2 &amp; 1/2 \\ <br /> 1/2 &amp; 1/2 <br /> \end{pmatrix}

and thus

<br /> S = - \mathrm{Tr} \left( \begin{pmatrix}<br /> 1/2 &amp; 1/2 \\ <br /> 1/2 &amp; 1/2 <br /> \end{pmatrix} \begin{pmatrix}<br /> -1 &amp; -1 \\ <br /> -1 &amp; -1 <br /> \end{pmatrix} \right) = - \mathrm{Tr} \begin{pmatrix}<br /> -1 &amp; -1 \\ <br /> -1 &amp; -1 <br /> \end{pmatrix} = 2<br />

Do I have to use the diagonal basis!? If not, what idiotic mistake am I making?
 
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To anyone wondering. No, basis does not matter. The error in the above reasoning is the assumption that the logarithm of a matrix is performed elementwise. This is true for diagonal matrices, but not general ones.

More concretely:

\log_2 \begin{pmatrix}<br /> 1/2 &amp; 1/2 \\ <br /> 1/2 &amp; 1/2 <br /> \end{pmatrix} \neq \begin{pmatrix}<br /> -1 &amp; -1 \\ <br /> -1 &amp; -1<br /> \end{pmatrix}<br />

Gah! Well, glad that's sorted.
 
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