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+# Math 401, Fall 2025: Thesis notes, R1, Non-commutative probability theory
+
+> Progress: 0/NaN=NaN% (denominator and enumerator may change)
+
+## Notations and definitions
+
+This part will cover the necessary notations and definitions for the remaining parts of the recollection.
+
+### Notations of Hilbert space
+
+A Hilbert space is a vector space equipped with an inner product.
+
+### Lipschitz function
+
+#### $\eta$-Lipschitz function
+
+Let $(X,\operatorname{dist}_X)$ and $(Y,\operatorname{dist}_Y)$ be two metric spaces. A function $f:X\to Y$ is said to be $\eta$-Lipschitz if there exists a constant $L\in \mathbb{R}$ such that
+
+$$
+\operatorname{dist}_Y(f(x),f(y))\leq L\operatorname{dist}_X(x,y)
+$$
+
+for all $x,y\in X$. And $\eta=\|f\|_{\operatorname{Lip}}=\inf_{L\in \mathbb{R}}L$.
+
+That basically means that the function $f$ should not change the distance between any two pairs of points in $X$ by more than a factor of $L$.
+
+### Operations on Hilbert space and Measurements
+
+Basic definitions
+
+### $SO(n)$
+
+The special orthogonal group $SO(n)$ is the set of all **distance preserving** linear transformations on $\mathbb{R}^n$.
+
+It is the group of all $n\times n$ orthogonal matrices ($A^T A=I_n$) on $\mathbb{R}^n$ with determinant $1$.
+
+$$
+SO(n)=\{A\in \mathbb{R}^{n\times n}: A^T A=I_n, \det(A)=1\}
+$$
+
+<details>
+<summary>Extensions</summary>
+
+In [The random Matrix Theory of the Classical Compact groups](https://case.edu/artsci/math/esmeckes/Haar_book.pdf), the author gives a more general definition of the Haar measure on the compact group $SO(n)$,
+
+$O(n)$ (the group of all $n\times n$ **orthogonal matrices** over $\mathbb{R}$),
+
+$$
+O(n)=\{A\in \mathbb{R}^{n\times n}: AA^T=A^T A=I_n\}
+$$
+
+$U(n)$ (the group of all $n\times n$ **unitary matrices** over $\mathbb{C}$), 
+
+$$
+U(n)=\{A\in \mathbb{C}^{n\times n}: A^*A=AA^*=I_n\}
+$$
+
+Recall that $A^*$ is the complex conjugate transpose of $A$.
+
+$SU(n)$ (the group of all $n\times n$ unitary matrices over $\mathbb{C}$ with determinant $1$), 
+
+$$
+SU(n)=\{A\in \mathbb{C}^{n\times n}: A^*A=AA^*=I_n, \det(A)=1\}
+$$
+
+$Sp(2n)$ (the group of all $2n\times 2n$ symplectic matrices over $\mathbb{C}$),
+
+$$
+Sp(2n)=\{U\in U(2n): U^T J U=UJU^T=J\}
+$$
+
+where $J=\begin{pmatrix}
+0 & I_n \\
+-I_n & 0
+\end{pmatrix}$ is the standard symplectic matrix.
+
+</details>
+
+### Haar measure
+
+Let $(SO(n), \| \cdot \|, \mu)$ be a metric measure space where $\| \cdot \|$ is the [Hilbert-Schmidt norm](https://notenextra.trance-0.com/Math401/Math401_T2#definition-of-hilbert-schmidt-norm) and $\mu$ is the measure function.
+
+The Haar measure on $SO(n)$ is the unique probability measure that is invariant under the action of $SO(n)$ on itself.
+
+That is also called _translation-invariant_.
+
+That is, fixing $B\in SO(n)$, $\forall A\in SO(n)$, $\mu(A\cdot B)=\mu(B\cdot A)=\mu(B)$.
+
+The Haar measure is the unique probability measure that is invariant under the action of $SO(n)$ on itself.
+
+_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._
+
+### Random sampling on the $\mathbb{C}P^n$
+
+Note that the space of pure state in bipartite system
+
+## Non-commutative probability theory
+
+### Partial trace and purification
+
+#### Partial trace
+
+Recall that the bipartite state of a quantum system is a linear operator on $\mathscr{H}=\mathscr{A}\otimes \mathscr{B}$, where $\mathscr{A}$ and $\mathscr{B}$ are finite-dimensional Hilbert spaces.
+
+##### Definition of partial trace for arbitrary linear operators
+
+Let $T$ be a linear operator on $\mathscr{H}=\mathscr{A}\otimes \mathscr{B}$, where $\mathscr{A}$ and $\mathscr{B}$ are finite-dimensional Hilbert spaces.
+
+An operator $T$ on $\mathscr{H}=\mathscr{A}\otimes \mathscr{B}$ can be written as (by the definition of [tensor product of linear operators](https://notenextra.trance-0.com/Math401/Math401_T2#tensor-products-of-linear-operators))
+
+$$
+T=\sum_{i=1}^n a_i A_i\otimes B_i
+$$
+
+where $A_i$ is a linear operator on $\mathscr{A}$ and $B_i$ is a linear operator on $\mathscr{B}$.
+
+The $\mathscr{B}$-partial trace of $T$ ($\operatorname{Tr}_{\mathscr{B}}(T):\mathcal{L}(\mathscr{A}\otimes \mathscr{B})\to \mathcal{L}(\mathscr{A})$) is the linear operator on $\mathscr{A}$ defined by
+
+$$
+\operatorname{Tr}_{\mathscr{B}}(T)=\sum_{i=1}^n a_i \operatorname{Tr}(B_i) A_i
+$$
+
+#### Definition of partial trace for density operators
+
+Let $\rho$ be a density operator in $\mathscr{H}_1\otimes\mathscr{H}_2$, the partial trace of $\rho$ over $\mathscr{H}_2$ is the density operator in $\mathscr{H}_1$ (reduced density operator for the subsystem $\mathscr{H}_1$) given by:
+
+$$
+\rho_1\coloneqq\operatorname{Tr}_2(\rho)
+$$
+
+<details>
+<summary>Examples</summary>
+
+Let $\rho=\frac{1}{\sqrt{2}}(|01\rangle+|10\rangle)$ be a density operator on $\mathscr{H}=\mathbb{C}^2\otimes \mathbb{C}^2$. 
+
+Expand the expression of $\rho$ in the basis of $\mathbb{C}^2\otimes\mathbb{C}^2$ using linear combination of basis vectors:
+
+$$
+\rho=\frac{1}{2}(|01\rangle\langle 01|+|01\rangle\langle 10|+|10\rangle\langle 01|+|10\rangle\langle 10|)
+$$
+
+Note $\operatorname{Tr}_2(|ab\rangle\langle cd|)=|a\rangle\langle c|\cdot \langle b|d\rangle$.
+
+Then the reduced density operator of the subsystem $\mathbb{C}^2$ in first qubit is, note the $\langle 0|0\rangle=\langle 1|1\rangle=1$ and $\langle 0|1\rangle=\langle 1|0\rangle=0$:
+
+$$
+\begin{aligned}
+\rho_1&=\operatorname{Tr}_2(\rho)\\
+&=\frac{1}{2}(\langle 1|1\rangle |0\rangle\langle 0|+\langle 0|1\rangle |0\rangle\langle 1|+\langle 1|0\rangle |1\rangle\langle 0|+\langle 0|0\rangle |1\rangle\langle 1|)\\
+&=\frac{1}{2}(|0\rangle\langle 0|+|1\rangle\langle 1|)\\
+&=\frac{1}{2}I
+\end{aligned}
+$$
+
+is a mixed state.
+
+</details>
+
+### Purification
+
+Let $\rho$ be any [state](https://notenextra.trance-0.com/Math401/Math401_T6#pure-states) (may not be pure) on the finite dimensional Hilbert space $\mathscr{H}$. then there exists a unit vector $w\in \mathscr{H}\otimes \mathscr{H}$ such that $\rho=\operatorname{Tr}_2(|w\rangle\langle w|)$ is a pure state.
+
+<details>
+<summary>Proof</summary>
+
+Let $(u_1,u_2,\cdots,u_n)$ be an orthonormal basis of $\mathscr{H}$ consisting of eigenvectors of $\rho$ for the eigenvalues $p_1,p_2,\cdots,p_n$. As $\rho$ is a states, $p_i\geq 0$ for all $i$ and $\sum_{i=1}^n p_i=1$.
+
+We can write $\rho$ as
+
+$$
+\rho=\sum_{i=1}^n p_i |u_i\rangle\langle u_i|
+$$
+
+Let $w=\sum_{i=1}^n \sqrt{p_i} u_i\otimes u_i$, note that $w$ is a unit vector (pure state). Then
+
+$$
+\begin{aligned}
+\operatorname{Tr}_2(|w\rangle\langle w|)&=\operatorname{Tr}_2(\sum_{i=1}^n \sum_{j=1}^n \sqrt{p_ip_j} |u_i\otimes u_i\rangle \langle u_j\otimes u_j|)\\
+&=\sum_{i=1}^n \sum_{j=1}^n \sqrt{p_ip_j} \operatorname{Tr}_2(|u_i\otimes u_i\rangle \langle u_j\otimes u_j|)\\
+&=\sum_{i=1}^n \sum_{j=1}^n \sqrt{p_ip_j} \langle u_i|u_j\rangle |u_i\rangle\langle u_i|\\
+&=\sum_{i=1}^n \sum_{j=1}^n \sqrt{p_ip_j} \delta_{ij} |u_i\rangle\langle u_i|\\
+&=\sum_{i=1}^n p_i |u_i\rangle\langle u_i|\\
+&=\rho
+\end{aligned}
+$$
+
+is a pure state.
+
+</details>
+
+## 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.
+
+Consider the function $f:S^{2N-1}\to \mathbb{R}$ defined by $f(x)=S(\rho(\psi))$, where $S(\cdot)$ is the von Neumann entropy. The Lipschitz constant of $f$ is $\sim \ln N$.