NoteNextra-origin/content/Math401/Extending_thesis/Math401_R1.md

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}

Examples of inner product

Let V=\mathbb{R}^n.

The dot product is defined by

\langle u,v\rangle=u_1v_1+u_2v_2+\cdots+u_nv_n

is an inner product.

---

Let V=L^2(\mathbb{R}, \lambda), where \lambda is the Lebesgue measure. f,g:\mathbb{R}\to \mathbb{C} are complex-valued square integrable functions.

The Hermitian inner product is defined by

\langle f,g\rangle=\int_\mathbb{R} \overline{f(x)}g(x) d\lambda(x)

is an inner product.

---

Let A,B be two linear transformation on \mathbb{R}^n.

The Hilbert-Schmidt inner product is defined by

\langle A,B\rangle=\operatorname{Tr}(A^*B)=\sum_{i=1}^n \sum_{j=1}^n \overline{a_{ij}}b_{ij}

is an inner product.

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

(v_1+v_2)\otimes w=(v_1\otimes w)+(v_2\otimes w)\text{ and } v\otimes (w_1+w_2)=(v\otimes w_1)+(v\otimes w_2)

and enables scalar multiplication by

\lambda (v\otimes w)=(\lambda v)\otimes w=v\otimes (\lambda w)

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

Tip

Note the difference between a linear functional and a linear map.

A generalized linear map is a function f:V\to W satisfying the condition

1. f(u+v)=f(u)+f(v)
2. f(\lambda v)=\lambda f(v)

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

(v\otimes w)(\psi,\phi)\coloneqq\psi(v)\phi(w)

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.

V\otimes W=\left\{\sum_{i=1}^n \sum_{j=1}^m a_{ij} \phi_i(v)\psi_j(w): \phi_i\in V', \psi_j\in W'\right\}

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

\langle v\otimes w, u\otimes x\rangle=\langle v,u\rangle_V\langle w,x\rangle_W

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.

Note

All those definitions and proofs can be found in Linear Algebra Done Right by Sheldon Axler.

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)

Examples of measure

The Borel measure on $\mathbb{R}$ is the collection of all closed, open, and half-open intervals with m(U)=\ell(U) for any open set U.

The Lebesgue measure on $\mathbb{R}$ is the collection of all Lebesgue measurable sets with m_i=\sup_{K\text{ closed},K\subseteq S}m(K) and m_e=\inf_{U\text{ open},S\subseteq U}m(U). and m(S)=m_e(S)=m_i(S) for any Lebesgue measurable set S.

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).

Examples of measurable space

Let \Omega be arbitrary set.

Let \mathscr{B}(\mathbb{C}) be the Borel sigma-algebra on \mathbb{C} generated from rectangles over complex plane with real number axes and \lambda be the Lebesgue measure associated with it.

Let \mathscr{F} be the set of square integrable, that is,

\int_\Omega |f(x)|^2 d\lambda(x)<\infty

complex-valued functions on \Omega, that is, f:\Omega\to \mathbb{C}.

Then the measurable space (\Omega, \mathscr{B}(\mathbb{C}), \lambda) is a measurable space. We usually denote this as L^2(\Omega, \mathscr{B}(\mathbb{C}), \lambda).

If \Omega=\mathbb{R}, then we denote such measurable space as L^2(\mathbb{R}, \lambda).

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

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

SO(n)=\{A\in \mathbb{R}^{n\times n}: A^\top A=I_n, \det(A)=1\}

Extensions

In The random Matrix Theory of the Classical Compact groups, 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^\top=A^\top 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^\top J U=UJU^\top=J\}

where $J=\begin{pmatrix} 0 & I_n \ -I_n & 0 \end{pmatrix}$ is the standard symplectic matrix.

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)

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)

Examples

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.

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.

Proof

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.

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.