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 Linear algebra

Definition of vector space

link to vector space

A vector space over $\mathbb{f}$ is a set $V$ along with two operators $v+w\in V$ for $v,w\in V$, and $\lambda \cdot v$ for $\lambda\in \mathbb{F}$ and $v\in V$ satisfying the following properties:

• Commutativity: $\forall v, w\in V,v+w=w+v$
• Associativity: $\forall u,v,w\in V,(u+v)+w=u+(v+w)$
• Existence of additive identity: $\exists 0\in V$ such that $\forall v\in V, 0+v=v$
• Existence of additive inverse: $\forall v\in V, \exists w \in V$ such that $v+w=0$
• Existence of multiplicative identity: $\exists 1 \in \mathbb{F}$ such that $\forall v\in V,1\cdot v=v$
• Distributive properties: $\forall v, w\in V$ and $\forall a,b\in \mathbb{F}$, $a\cdot(v+w)=a\cdot v+ a\cdot w$ and $(a+b)\cdot v=a\cdot v+b\cdot v$

Definition of inner product

link to inner product

An inner product is a bilinear function $\langle,\rangle:V\times V\to \mathbb{F}$ satisfying the following properties:

• Positivity: $\langle v,v\rangle\geq 0$
• Definiteness: $\langle v,v\rangle=0\iff v=0$
• Additivity: $\langle u+v,w\rangle=\langle u,w\rangle+\langle v,w\rangle$
• Homogeneity: $\langle \lambda u, v\rangle=\lambda\langle u,v\rangle$
• Conjugate symmetry: $\langle u,v\rangle=\overline{\langle v,u\rangle}$

Definition of inner product space

A inner product space is a vector space equipped with an inner product.

Definition of completeness

link to completeness

Note that every inner product space is a metric space.

Let $X$ be a metric space. We say $X$ is complete if every Cauchy sequence (that is, a sequence such that $\forall \epsilon>0, \exists N$ such that $\forall m,n\geq N, d(p_m,p_n)<\epsilon$) in $X$ converges.

Definition of Hilbert space

A Hilbert space is a complete inner product space.

Motivation of Tensor product

Recall from the traditional notation of product space of two vector spaces $V$ and $W$, that is, $V\times W$, is the set of all ordered pairs $(v,w)$ where $v\in V$ and $w\in W$.

The space has dimension $\dim V+\dim W$.

We want to define a vector space with notation of multiplication of two vectors from different vector spaces.

That is

and enables scalar multiplication by

And we wish to build a way associates the basis of $V$ and $W$ to the basis of $V\otimes W$. That makes the tensor product a vector space with dimension $\dim V\times \dim W$.

Definition of linear functional

A linear functional is a linear map from $V$ to $\mathbb{F}$.

Definition of bilinear functional

A bilinear functional is a bilinear function $\beta:V\times W\to \mathbb{F}$ satisfying the condition that $v\to \beta(v,w)$ is a linear functional for all $w\in W$ and $w\to \beta(v,w)$ is a linear functional for all $v\in V$.

The vector space of all bilinear functionals is denoted by $\mathcal{B}(V,W)$.

Definition of tensor product

Let $V,W$ be two vector spaces.

Let $V'$ and $W'$ be the dual spaces of $V$ and $W$, respectively, that is $V'=\{\psi:V\to \mathbb{F}\}$ and $W'=\{\phi:W\to \mathbb{F}\}$, $\psi, \phi$ are linear functionals.

The tensor product of vectors $v\in V$ and $w\in W$ is the bilinear functional defined by $\forall (\psi,\phi)\in V'\times W'$ given by the notation

The tensor product of two vector spaces $V$ and $W$ is the vector space $\mathcal{B}(V',W')$

Notice that the basis of such vector space is the linear combination of the basis of $V'$ and $W'$, that is, if $\{e_i\}$ is the basis of $V'$ and $\{f_j\}$ is the basis of $W'$, then $\{e_i\otimes f_j\}$ is the basis of $\mathcal{B}(V',W')$.

That is, every element of $\mathcal{B}(V',W')$ can be written as a linear combination of the basis.

Since $\{e_i\}$ and $\{f_j\}$ are bases of $V'$ and $W'$, respectively, then we can always find a set of linear functionals $\{\phi_i\}$ and $\{\psi_j\}$ such that $\phi_i(e_j)=\delta_{ij}$ and $\psi_j(f_i)=\delta_{ij}$.

Here $\delta_{ij}=\begin{cases} 1 & \text{if } i=j \\ 0 & \text{otherwise} \end{cases}$ is the Kronecker delta.

Note that $\sum_{i=1}^n \sum_{j=1}^m a_{ij} \phi_i(v)\psi_j(w)$ is a bilinear functional that maps $V'\times W'$ to $\mathbb{F}$.

This enables basis free construction of vector spaces with proper multiplication and scalar multiplication.

This vector space is equipped with the unique inner product $\langle v\otimes w, u\otimes x\rangle_{V\otimes W}$ defined by

In practice, we ignore the subscript of the vector space and just write $\langle v\otimes w, u\otimes x\rangle=\langle v,u\rangle\langle w,x\rangle$.

Notations in measure theory

Definition of Sigma algebra

link to measure theory

A collection of sets $\mathcal{A}$ is called a sigma-algebra if it satisfies the following properties:

1. $\emptyset \in \mathcal{A}$
2. If $\{A_j\}_{j=1}^\infty \subset \mathcal{A}$, then $\bigcup_{j=1}^\infty A_j \in \mathcal{A}$
3. If $A \in \mathcal{A}$, then $A^c \in \mathcal{A}$

Definition of Measure

A measure is a function $v:\mathcal{A}\to \mathbb{R}$ satisfying the following properties:

1. $v(\emptyset)=0$
2. If $\{A_j\}_{j=1}^\infty \subset \mathcal{A}$ are pairwise disjoint, then $v(\bigcup_{j=1}^\infty A_j)=\sum_{j=1}^\infty v(A_j)$ (countable additivity)
3. If $A\in \mathcal{A}$, then $v(A)\geq 0$ (non-negativity)

Definition of Probability measure

Let $\mathscr{F}$ be a sigma-algebra on a set $\Omega$. A probability measure is a function $P:\mathscr{F}\to [0,1]$ satisfying the following properties:

1. $P(\Omega)=1$
2. $P$ is a measure on $\mathscr{F}$

Definition of Measurable space

A measurable space is a pair $(X, \mathscr{B}, v)$, where $X$ is a set and $\mathscr{B}$ is a sigma-algebra on $X$.

In some literatures, $\mathscr{B}$ is ignored and we only denote it as $(X, v)$.

Probability space

A probability space is a triple $(\Omega, \mathscr{F}, P)$, where $\Omega$ is a set, $\mathscr{F}$ is a sigma-algebra on $\Omega$, and $P$ is a probability measure on $\mathscr{F}$.

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

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 ($A^\top A=I_n$) on $\mathbb{R}^n$ with determinant $1$.

Haar measure

Let $(SO(n), \| \cdot \|, \mu)$ be a metric measure space where $\| \cdot \|$ is the 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

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

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 )

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

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:

Purification

Let $\rho$ be any state  (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.

Drawing the connection between the space $S^{2n+1}$, $\mathbb{C}P^n$, 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 $\sim \ln N$.