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Math 401, Fall 2025: Thesis notes, R1, Non-commutative probability theory
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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\}
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^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.
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}, 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.