Update week5.do.txt

Author: mhjensen

doc/src/week5/week5.do.txt

@@ -310,21 +310,29 @@ just Shannon entropy, before we move over to a quantum mechanical way
 to define the entropy based on the density matrices discussed earlier.
 
 We define a set of random variables $X=\{x_0,x_1,\dots,x_{n-1}\}$ with probability for an outcome $x\in X$ given by $p_X(x)$, the
-information entropy is defined as
+classical information entropy is defined as
 !bt
 \[
 S=-\sum_{x\in X}p_X(x)\log_2{p_X(x)}.
 \]
 !et
+Why this expression? What does it mean?
 
+!split
+===== Mathematics of entropy =====
+What is the basic idea of the entropy as it is used in information theory (we leave out the standard description from statistical physics here)?
+
+We want to have a measure of unlikelihod. Consider a simple binary system, with two outcomes, true and false with true given by a probability $p$ and false given by $1-p$. Since $p$ represents a probability 
 !split
 ===== Von Neumann entropy =====
 
+Using the density matrix $\rho$, we define the quantum mechanical equivalent of the classical entropy as 
 !bt
 \[
 S=-\mathrm{Tr}[\rho\log_2{\rho}].
 \]
 !et
+This is the so-called Von Neumann entropy. How did we arrive at this expression?