update?

content/Math401/Math401_P1.md

@@ -101,3 +101,4 @@ $B$ decodes the result and obtains the 2 classical bits sent by $A$.
 
 ### Multipartite entanglement
 
+> The role of the paper in Physics can be found in (15.86) on book Geometry of Quantum states.

content/Math401/Math401_P1_1.md

@@ -145,4 +145,8 @@ QED
 
 ## Drawing the connection between the space $S^{2n+1}$, $CP^n$, and $\mathbb{R}$
 
-##
+A pure quantum state of size $N$ can be identified with a Hopf circle on the sphere $S^{2N-1}$.
+
+A random pure state $|\psi\rangle$ of a bipartite $N\times K$ system such that $K\geq N\geq 3$.
+
+The partial trace of such system produces a mixed state $\rho(\psi)=\operatorname{Tr}_K(|\psi\rangle\langle \psi|)$, with induced measure $\mu_K$. When $K=N$, the induced measure $\mu_K$ is the Hilbert-Schmidt measure.

content/Math401/Math401_P1_2.md

@@ -66,10 +66,6 @@ The Haar measure is the unique probability measure that is invariant under the a
 
 _The existence and uniqueness of the Haar measure is a theorem in compact lie group theory. For this research topic, we will not prove it._
 
-### Maxwell-Boltzmann distribution and projection of high-dimensional sphere
-
-
-
 ### Random sampling on the $CP^n$
 
 ## Statement
@@ -98,4 +94,6 @@ $$
 
 - [Page's conjecture](https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.71.1291)
 
-- [Page's conjecture simple proof](https://journals.aps.org/pre/pdf/10.1103/PhysRevE.52.5653)
+- [Page's conjecture simple proof](https://journals.aps.org/pre/pdf/10.1103/PhysRevE.52.5653)
+
+- [Geometry of quantum states an introduction to quantum entanglement second edition](https://www.cambridge.org/core/books/geometry-of-quantum-states/46B62FE3F9DA6E0B4EDDAE653F61ED8C)

content/Math401/Math401_T2.md

@@ -257,7 +257,7 @@ $$
 \text{Range}(A)=\{w\in \mathscr{W}: \exists v\in \mathscr{V}, Av=w\}
 $$
 
-### Dual spaces and adjoints of linear maps\
+### Dual spaces and adjoints of linear maps
 
 #### Definition of linear map
 
@@ -321,7 +321,7 @@ $$
 \langle v|w\rangle
 $$
 
-is the inner product of $v$ and $w$.
+is the inner product of $v$ and $w$. That is, $\langle v|w\rangle: \mathscr{H}\to \mathbb{C}$ is a linear functional satisfying the property of inner product.
 
 $$
 |v\rangle
@@ -606,10 +606,18 @@ for all $u_i\in \mathscr{H}_1$ and $v_i\in \mathscr{H}_2$.
 
 Such tensor product of linear operators is well defined.
 
+<details>
+<summary>Proof</summary>
+
 If $\sum_{i=1}^n a_i u_i\otimes v_i=\sum_{j=1}^m b_j u_j\otimes v_j$, then $a_i=b_j$ for all $i=1,2,\cdots,n$ and $j=1,2,\cdots,m$.
 
 Then $\sum_{i=1}^n a_i T_1(u_i)\otimes T_2(v_i)=\sum_{j=1}^m b_j T_1(u_j)\otimes T_2(v_j)$.
 
+QED
+</details>
+
+An example of 
+
 #### Tensor product of linear operators on Hilbert spaces
 
 Let $T_1$ be a linear operator on $\mathscr{H}_1$ and $T_2$ be a linear operator on $\mathscr{H}_2$, where $\mathscr{H}_1$ and $\mathscr{H}_2$ are finite-dimensional Hilbert spaces. The tensor product of $T_1$ and $T_2$ (denoted by $T_1\otimes T_2$) on $\mathscr{H}_1\otimes \mathscr{H}_2$, such that **on decomposable elements** is defined by