What is the definition of log(P) in the von neumann entropy formula?

[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.

 

[humanino]

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

 

[humanino]

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.

right. Usually easier by the way !

 

[dextercioby]

The definition is really

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

,quite similar to Gibbs' entropy.

Daniel.