Update week5.do.txt

Author: mhjensen

doc/src/week5/week5.do.txt

@@ -422,15 +422,36 @@ Let us assume we are studying specific system $A$. The density matrix,
 using an ONB basis $\psi_j$ is defined as
 !bt
 \[
-\rho_A=\sum_j\lambda_j\vert \psi_j\rangle\langle \psi_j\vert=\sum_j p_j\vert \psi_j\rangle\langle \psi_j\vert,
+\bm{\rho}_A=\sum_j\lambda_j\vert \psi_j\rangle\langle \psi_j\vert=\sum_j p_j\vert \psi_j\rangle\langle \psi_j\vert,
 \]
 !et
 and the eigenvalues $\lambda_j$ are the probabilities (or overlap coefficients) of being in a
 specific state $\psi_j$. 
 
+Note that the density matrix/operator is a semi-positive definite matrix.
+
 !split
 ===== Linking with a new expression for the entropy =====
 
+Using a unitary transformation $\bm{U}$ we can transform the density
+matrix into a diagonal matrix $\bm{D}_a$ where the eigenvalues are the
+above mentioned probabilities (overlap coefficients squared). We can then define
+!bt
+\[
+\bm{D}_A=\bm{U}^{\dagger}\bm{\rho}_A\bm{U}=\begin{bmatrix} p_0 & 0 & 0 & \dots & 0\\
+ 0 & p_1 & 0 & \dots & 0\\
+ 0 & 0 & p_2 & \dots & 0\\
+\dots & \dots & \dots & \dots & \dots\\
+\dots & \dots & \dots & \dots & \dots\\
+\dots & \dots & \dots & \dots & \dots\\
+0 & 0 & \dots & 0 & p_{n-1}\\
+\end{bmatrix}.
+\]
+!et
+
+!split
+===== Von Neumann entropy =====
+
 
 
 !split