Update Math401_T4.md

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@@ -36,7 +36,7 @@ A pair of state and effect determines a probability $E_i(P)=p(\omega_i|P)$. By t
 
 **Registration:** The impression left on the glass pace by the condensed silver atoms.
 
-## Finite probability spaces in the language of Hilbert space and operators
+## Section 3: Finite probability spaces in the language of Hilbert space and operators
 
 > Superposition is a linear combination of two or more states.
 
@@ -44,3 +44,57 @@ A quantum coin can be represented mathematically by linear combination of $|0\ra
 
 > For the rest of the material, we shall take the $\mathscr{H}$ to be vector space over $\mathbb{C}$.
 
+### Rewrite the language of probability
+
+To each event $A\in \Omega$, we associate the operator on $\mathscr{H}$ of multiplication by the indicator function $M_{\mathbb{I}_A}:f\mapsto \mathbb{I}_A f$ that projects onto the subspace of $\mathscr{H}$ corresponding to the event $A$.
+
+$$
+P_A=\sum_{k=1}^n a_k|k\rangle\langle k|
+$$
+
+where $a_k\in\{0,1\}$, and $a_k=1$ if and only if $k\in A$. Note that $P_A^*=P_A$ and $P_A^2=P_A$.
+
+#### Density operator
+
+Let $(p_1,p_2,\cdots,p_n)$ be a probability distribution on $X$, where $p_k\geq 0$ and $\sum_{k=1}^n p_k=1$. The density operator $\rho$ is defined by
+
+$$
+\rho=\sum_{k=1}^n p_k|k\rangle\langle k|
+$$
+
+The probability of event $A$ relative to the probability distribution $(p_1,p_2,\cdots,p_n)$ becomes the trace of the product of $\rho$ and $P_A$.
+
+$$
+\operatorname{Prob}_\rho(A)=\text{Tr}(\rho P_A)
+$$
+
+#### Random variables
+
+A random variable is a function $f:X\to\mathbb{R}$. It can also be written in operator form:
+
+$$
+F=\sum_{k=1}^n f(k)P_{\{k\}}
+$$
+
+The expectation of $f$ relative to the probability distribution $(p_1,p_2,\cdots,p_n)$ is given by
+
+$$
+\mathbb{E}_\rho(f)=\sum_{k=1}^n p_k f(k)=\operatorname{Tr}(\rho F)
+$$
+
+Note, by our definition of the operator $F,P_A,\rho$ (all diagonal operators) commute among themselves, which does not hold in general quantum theory.
+
+## Section 4: Why we need generalized probability theory to study quantum systems
+
+Story of light polarization.
+
+### Classical picture of light polarization and Bell's inequality
+
+> An interesting story will be presented here.
+
+## Section 5: The quantum probability theory
+
+Let $\mathscr{H}$ be a Hilbert space. $\mathscr{H}$ consists of complex-valued functions on a finite set $\Omega=\{1,2,\cdots,n\}$. and that the functions $(e_1,e_2,\cdots,e_n)$ form an orthonormal basis of $\mathscr{H}$. We use Dirac notation $|k\rangle$ to denote the basis vector $e_k$.
+
+In classical settings, multiplication operators is now be the full space of bounded linear operators on $\mathscr{H}$. (Denoted by $\mathscr{B}(\mathscr{H})$)
+