How Is Entropy Calculated for a Density Matrix with Eigenvalues 0 and 1?

[LagrangeEuler]

Homework Statement

Calculate entropy for density matrix with eigenvalues $0$ and $1$.

Homework Equations

$S=-\lambda_1 \ln \lambda_1-\lambda_2 \ln \lambda_2$
where $\lambda_1$ and $\lambda_2$ are eigenvalues of density matrix.

The Attempt at a Solution

How to calculate this when $\ln 0$ is not defined?

 

[Mandelbroth]

Homework Statement

Calculate entropy for density matrix with eigenvalues $0$ and $1$.

Homework Equations

$S=-\lambda_1 \ln \lambda_1-\lambda_2 \ln \lambda_2$
where $\lambda_1$ and $\lambda_2$ are eigenvalues of density matrix.

The Attempt at a Solution

How to calculate this when $\ln 0$ is not defined?

I'm no expert, but...

What is the value of $\displaystyle \lim_{\lambda_1\rightarrow 0}\left[\lambda_1\ln{\lambda_1}\right]$?

I think that is the only way to get a numerical answer here: make it approach what you want.

 

[tannerbk]

For a density matrix ρ with eigenvalues only 0 and 1, we have $ρ = ρ^{2}$. This is true only for pure states and thus we know the Von Neumann entropy must be zero. To calculate it numerically I would guess the approach Mandelbroth suggested is valid.

 

[LagrangeEuler]

What is interpretation of that. For pure state entropy is zero. Why?
$-1\ln 1-\lim_{\lambda \rightarrow 0}\lambda \ln \lambda=-\lim_{\lambda \rightarrow 0}\lambda \ln \lambda$
How to calculate this limit?

 

[tannerbk]

What is interpretation of that. For pure state entropy is zero. Why?
$-1\ln 1-\lim_{\lambda \rightarrow 0}\lambda \ln \lambda=-\lim_{\lambda \rightarrow 0}\lambda \ln \lambda$
How to calculate this limit?

To calculate the limit let $t = 1/x$. Then you have $$\lim_{t\to\infty}=\frac{log(1/t)}{t} = \frac{\infty}{\infty}.$$ Then use L'Hospital's Rule and you will get the answer. Otherwise, you could just type it into wolfram alpha.

 

[LagrangeEuler]

Tnx a lot! And physically why entropy of pure state is zero?

 

[tannerbk]

It is easy to see mathematically why the entropy of a pure state is zero. However, why its true physically seems a much harder question, one I'm not sure I know how to answer.

 

[Mute]

Entropy is in some sense a measure of our uncertainty about the state a system. If a system is in a mixed state, is our uncertainty big or small? What about when it is in a pure state?

 

[tannerbk]

Entropy is in some sense a measure of our uncertainty about the state a system. If a system is in a mixed state, is our uncertainty big or small? What about when it is in a pure state?

Pure state is minimum uncertainty so it makes sense Entropy would be zero.