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

Eigenentity
Messages
7
Reaction score
0
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?
 
Physics news on Phys.org


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.
 
Last edited:
Hi, I had an exam and I completely messed up a problem. Especially one part which was necessary for the rest of the problem. Basically, I have a wormhole metric: $$(ds)^2 = -(dt)^2 + (dr)^2 + (r^2 + b^2)( (d\theta)^2 + sin^2 \theta (d\phi)^2 )$$ Where ##b=1## with an orbit only in the equatorial plane. We also know from the question that the orbit must satisfy this relationship: $$\varepsilon = \frac{1}{2} (\frac{dr}{d\tau})^2 + V_{eff}(r)$$ Ultimately, I was tasked to find the initial...
The value of H equals ## 10^{3}## in natural units, According to : https://en.wikipedia.org/wiki/Natural_units, ## t \sim 10^{-21} sec = 10^{21} Hz ##, and since ## \text{GeV} \sim 10^{24} \text{Hz } ##, ## GeV \sim 10^{24} \times 10^{-21} = 10^3 ## in natural units. So is this conversion correct? Also in the above formula, can I convert H to that natural units , since it’s a constant, while keeping k in Hz ?

Similar threads

Back
Top