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Eigenentity
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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
[tex]S = - \mathrm{Tr}(\rho \log_2{\rho})[/tex]
for the pure state
[tex]| \psi \rangle = \left(|0\rangle + |1\rangle\right)/ \sqrt 2[/tex]
Now, clearly the simplest thing to do is to express [tex]|\psi\rangle[/tex] in the [tex]|+\rangle,|-\rangle[/tex] basis, where it's clear that
[tex]\rho = |+\rangle\langle+|[/tex]
In this basis, [tex]S = 0[/tex] 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 [tex]|0\rangle,|1\rangle[/tex] basis,
[tex]\rho = \begin{pmatrix}
1/2 & 1/2 \\
1/2 & 1/2
\end{pmatrix}[/tex]
and thus
[tex]
S = - \mathrm{Tr} \left( \begin{pmatrix}
1/2 & 1/2 \\
1/2 & 1/2
\end{pmatrix} \begin{pmatrix}
-1 & -1 \\
-1 & -1
\end{pmatrix} \right) = - \mathrm{Tr} \begin{pmatrix}
-1 & -1 \\
-1 & -1
\end{pmatrix} = 2
[/tex]
Do I have to use the diagonal basis!? If not, what idiotic mistake am I making?
I'm very confused by why I'm unable to correctly compute the von Neumann entropy
[tex]S = - \mathrm{Tr}(\rho \log_2{\rho})[/tex]
for the pure state
[tex]| \psi \rangle = \left(|0\rangle + |1\rangle\right)/ \sqrt 2[/tex]
Now, clearly the simplest thing to do is to express [tex]|\psi\rangle[/tex] in the [tex]|+\rangle,|-\rangle[/tex] basis, where it's clear that
[tex]\rho = |+\rangle\langle+|[/tex]
In this basis, [tex]S = 0[/tex] 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 [tex]|0\rangle,|1\rangle[/tex] basis,
[tex]\rho = \begin{pmatrix}
1/2 & 1/2 \\
1/2 & 1/2
\end{pmatrix}[/tex]
and thus
[tex]
S = - \mathrm{Tr} \left( \begin{pmatrix}
1/2 & 1/2 \\
1/2 & 1/2
\end{pmatrix} \begin{pmatrix}
-1 & -1 \\
-1 & -1
\end{pmatrix} \right) = - \mathrm{Tr} \begin{pmatrix}
-1 & -1 \\
-1 & -1
\end{pmatrix} = 2
[/tex]
Do I have to use the diagonal basis!? If not, what idiotic mistake am I making?