diff --git a/content/Math401/Extending_thesis/Math401_R1.md b/content/Math401/Extending_thesis/Math401_R1.md
index 44426f6..e50f83c 100644
--- a/content/Math401/Extending_thesis/Math401_R1.md
+++ b/content/Math401/Extending_thesis/Math401_R1.md
@@ -96,6 +96,14 @@ Note that the space of pure state in bipartite system
 
 ## Non-commutative probability theory
 
+### Pure state and mixed state
+
+A pure state is a state that is represented by a unit vector in $\mathscr{H}^{\otimes N}$.
+
+> As analogy, a pure state is the basis element of the vector space, a mixed state is a linear combination of basis elements.
+
+A mixed state is a state that is represented by a density operator (linear combination of pure states) in $\mathscr{H}^{\otimes N}$.
+
 ### Partial trace and purification
 
 #### Partial trace
diff --git a/content/Math401/Extending_thesis/Math401_S1.md b/content/Math401/Extending_thesis/Math401_S1.md
index ae08885..33e1fe4 100644
--- a/content/Math401/Extending_thesis/Math401_S1.md
+++ b/content/Math401/Extending_thesis/Math401_S1.md
@@ -12,3 +12,9 @@ $$
 \mathbb{C}P^n=\frac{\mathbb{C}^{n+1}\setminus\{0\}}{\sim}
 $$
 
+By this nature of ray-like properties, we can also describe the complex projective space as follows (in the math of QT, lecture 5)
+
+$$
+\mathbb{C}P^n=\left\{z=(z_0,z_1,\cdots,z_n)\in\mathbb{C}^{n+1}:|z_1|^2+\cdots+|z_n|^2=1\right\}/\sim
+$$
+