# Von neumann entropy, log(P) ?

1. Mar 29, 2005

### trosten

Let P be a density matrix. Then the von neumann entropy is defined as
S(P) = -tr(P*log(P))

But how is log(P) defined ?

--edit--
found the answer. the trace is independent of representation so that if P is diagonalized with eigenvalues {k} then S(P) = H({k}) where H is the shannon entropy.

Last edited: Mar 29, 2005
2. Mar 29, 2005

### humanino

$$ln(P) = -\sum_{n=1}^\infty \frac{1}{n}(I-A)^n$$

3. Mar 29, 2005

### humanino

right. Usually easier by the way !

4. Mar 29, 2005

### dextercioby

The definition is really

$$S:=-k\langle \ln\hat{\rho}\rangle_{\hat{\rho}}$$

,quite similar to Gibbs' entropy.

Daniel.