From d8e5e34e289f6414961f98bbb39b5c66e59a674c Mon Sep 17 00:00:00 2001 From: Zheyuan Wu <60459821+Trance-0@users.noreply.github.com> Date: Sat, 12 Jul 2025 17:00:42 -0500 Subject: [PATCH] updates --- content/CSE442T/CSE442T_L1.md | 2 +- content/CSE442T/CSE442T_L15.md | 2 +- content/CSE442T/CSE442T_L18.md | 2 +- content/CSE442T/CSE442T_L19.md | 2 +- content/CSE442T/CSE442T_L2.md | 2 +- content/CSE442T/CSE442T_L20.md | 2 +- content/CSE442T/CSE442T_L21.md | 2 +- content/CSE442T/CSE442T_L22.md | 2 +- content/CSE442T/CSE442T_L23.md | 2 +- content/CSE442T/CSE442T_L24.md | 2 +- content/CSE442T/CSE442T_L3.md | 2 +- content/CSE442T/CSE442T_L4.md | 2 +- content/Math401/Math401_P1_1.md | 33 +++++++++++++++++++-- content/Math401/Math401_P1_3.md | 15 ++++++++-- content/Math401/Math401_T2.md | 47 +++++++++++++++++++++++++----- content/contact.md | 4 ++- favicon.ico => public/favicon.ico | Bin favicon.svg => public/favicon.svg | 0 18 files changed, 99 insertions(+), 24 deletions(-) rename favicon.ico => public/favicon.ico (100%) rename favicon.svg => public/favicon.svg (100%) diff --git a/content/CSE442T/CSE442T_L1.md b/content/CSE442T/CSE442T_L1.md index 8db27c2..40280ef 100644 --- a/content/CSE442T/CSE442T_L1.md +++ b/content/CSE442T/CSE442T_L1.md @@ -1,4 +1,4 @@ -# Lecture 1 +# CSE442T Lecture 1 ## Chapter 1: Introduction diff --git a/content/CSE442T/CSE442T_L15.md b/content/CSE442T/CSE442T_L15.md index d5673bc..560a7b1 100644 --- a/content/CSE442T/CSE442T_L15.md +++ b/content/CSE442T/CSE442T_L15.md @@ -1,4 +1,4 @@ -# Lecture 15 +# CSE442T Lecture 15 ## Chapter 3: Indistinguishability and Pseudorandomness diff --git a/content/CSE442T/CSE442T_L18.md b/content/CSE442T/CSE442T_L18.md index 93c4555..ff7e381 100644 --- a/content/CSE442T/CSE442T_L18.md +++ b/content/CSE442T/CSE442T_L18.md @@ -1,4 +1,4 @@ -# Lecture 18 +# CSE442T Lecture 18 ## Chapter 5: Authentication diff --git a/content/CSE442T/CSE442T_L19.md b/content/CSE442T/CSE442T_L19.md index 3c4a94f..5c470f8 100644 --- a/content/CSE442T/CSE442T_L19.md +++ b/content/CSE442T/CSE442T_L19.md @@ -1,4 +1,4 @@ -# Lecture 19 +# CSE442T Lecture 19 ## Chapter 5: Authentication diff --git a/content/CSE442T/CSE442T_L2.md b/content/CSE442T/CSE442T_L2.md index 52fa65f..ee97328 100644 --- a/content/CSE442T/CSE442T_L2.md +++ b/content/CSE442T/CSE442T_L2.md @@ -1,4 +1,4 @@ -# Lecture 2 +# CSE442T Lecture 2 ## Probability review diff --git a/content/CSE442T/CSE442T_L20.md b/content/CSE442T/CSE442T_L20.md index 34585f9..84a6080 100644 --- a/content/CSE442T/CSE442T_L20.md +++ b/content/CSE442T/CSE442T_L20.md @@ -1,4 +1,4 @@ -# Lecture 20 +# CSE442T Lecture 20 ## Chapter 5: Authentication diff --git a/content/CSE442T/CSE442T_L21.md b/content/CSE442T/CSE442T_L21.md index eb4fc89..6f8467a 100644 --- a/content/CSE442T/CSE442T_L21.md +++ b/content/CSE442T/CSE442T_L21.md @@ -1,4 +1,4 @@ -# Lecture 21 +# CSE442T Lecture 21 ## Chapter 5: Authentication diff --git a/content/CSE442T/CSE442T_L22.md b/content/CSE442T/CSE442T_L22.md index d286fda..5ff62d8 100644 --- a/content/CSE442T/CSE442T_L22.md +++ b/content/CSE442T/CSE442T_L22.md @@ -1,4 +1,4 @@ -# Lecture 22 +# CSE442T Lecture 22 ## Chapter 7: Composability diff --git a/content/CSE442T/CSE442T_L23.md b/content/CSE442T/CSE442T_L23.md index 1fc4263..77f290b 100644 --- a/content/CSE442T/CSE442T_L23.md +++ b/content/CSE442T/CSE442T_L23.md @@ -1,4 +1,4 @@ -# Lecture 23 +# CSE442T Lecture 23 ## Chapter 7: Composability diff --git a/content/CSE442T/CSE442T_L24.md b/content/CSE442T/CSE442T_L24.md index 6dbf423..fe3b3b8 100644 --- a/content/CSE442T/CSE442T_L24.md +++ b/content/CSE442T/CSE442T_L24.md @@ -1,4 +1,4 @@ -# Lecture 24 +# CSE442T Lecture 24 ## Chapter 7: Composability diff --git a/content/CSE442T/CSE442T_L3.md b/content/CSE442T/CSE442T_L3.md index 8c00221..784a019 100644 --- a/content/CSE442T/CSE442T_L3.md +++ b/content/CSE442T/CSE442T_L3.md @@ -1,4 +1,4 @@ -# Lecture 3 +# CSE442T Lecture 3 All algorithms $C(x)\to y$, $x,y\in \{0,1\}^*$ diff --git a/content/CSE442T/CSE442T_L4.md b/content/CSE442T/CSE442T_L4.md index d904f0b..ca9e93b 100644 --- a/content/CSE442T/CSE442T_L4.md +++ b/content/CSE442T/CSE442T_L4.md @@ -1,4 +1,4 @@ -# Lecture 4 +# CSE442T Lecture 4 ## Recap diff --git a/content/Math401/Math401_P1_1.md b/content/Math401/Math401_P1_1.md index f58611b..511f5d0 100644 --- a/content/Math401/Math401_P1_1.md +++ b/content/Math401/Math401_P1_1.md @@ -52,9 +52,38 @@ Practically speaking: ## 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 + +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 + +$$ +\operatorname{Tr}_{\mathscr{B}}(T)=\sum_{i=1}^n a_i \operatorname{Tr}(B_i) A_i +$$ + +### 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. + +
+Proof + + + +
+ ## MM space diff --git a/content/Math401/Math401_P1_3.md b/content/Math401/Math401_P1_3.md index 42562f1..77635c7 100644 --- a/content/Math401/Math401_P1_3.md +++ b/content/Math401/Math401_P1_3.md @@ -1,8 +1,9 @@ # Math 401, Paper 1, Side note 3: Levy's concentration theorem -## Levy's concentration theorem in _High-dimensional probability_ by Roman Vershynin +## Basic definitions + +### Lipschitz function -### Levy's concentration theorem (Vershynin's version) #### $\eta$-Lipschitz function @@ -16,6 +17,16 @@ 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 $L$. +### Sub-Gaussian concentration + +### Random sampling on the $CP^n$ + + + +## Levy's concentration theorem in _High-dimensional probability_ by Roman Vershynin + +### Levy's concentration theorem (Vershynin's version) + > This theorem is exactly the 5.1.4 on the _High-dimensional probability_ by Roman Vershynin. #### Isoperimetric inequality on $\mathbb{R}^n$ diff --git a/content/Math401/Math401_T2.md b/content/Math401/Math401_T2.md index 8baa91c..23b2713 100644 --- a/content/Math401/Math401_T2.md +++ b/content/Math401/Math401_T2.md @@ -387,13 +387,13 @@ $$ This is equivalent to the sum of the diagonal elements of $T$. -> Check the rest of the section defining the partial trace by viewing the tensor product section first. +> Note, I changed the order of the definitions for the trace to pack similar concepts together. Check the rest of the section defining the partial trace by viewing the [tensor product section](https://notenextra.trance-0.com/Math401/Math401_T2#tensor-products-of-finite-dimensional-hilbert-spaces) first, and return to this section after reading the tensor product of linear operators. #### Definition of partial trace Let $T$ be a linear operator on $\mathscr{H}=\mathscr{A}\otimes \mathscr{B}$, where $\mathscr{A}$ and $\mathscr{B}$ are finite-dimensional Hilbert spaces. -An operator $T$ on $\mathscr{H}=\mathscr{A}\otimes \mathscr{B}$ can be written as +An operator $T$ on $\mathscr{H}=\mathscr{A}\otimes \mathscr{B}$ can be written as (by the definition of [tensor product of linear operators](https://notenextra.trance-0.com/Math401/Math401_T2#tensor-products-of-linear-operators)) $$ T=\sum_{i=1}^n a_i A_i\otimes B_i @@ -401,7 +401,7 @@ $$ where $A_i$ is a linear operator on $\mathscr{A}$ and $B_i$ is a linear operator on $\mathscr{B}$. -The partial trace of $T$ is the linear operator on $\mathscr{B}$ defined by +The $\mathscr{B}$-partial trace of $T$ ($\operatorname{Tr}_{\mathscr{B}}(T):\mathcal{L}(\mathscr{A}\otimes \mathscr{B})\to \mathcal{L}(\mathscr{A})$) is the linear operator on $\mathscr{A}$ defined by $$ \operatorname{Tr}_{\mathscr{B}}(T)=\sum_{i=1}^n a_i \operatorname{Tr}(B_i) A_i @@ -497,6 +497,20 @@ $$ f(x_j)=\sum_{a\in X_j} f(a)\epsilon_{a}^{(j)}(x_j) $$ +
+Proof + +Note that a function is a map for all elements in the domain. + +For each $a\in X_j$, $\epsilon_{a}^{(j)}(x_j)=1$ if $x_j=a$ and $0$ otherwise. So + +$$ +f(x_j)=\sum_{a\in X_j} f(a)\epsilon_{a}^{(j)}(x_j)=f(x_j) +$$ + +QED. + +
Now, let $a=(a_1,a_2,\cdots,a_n)$ be a vector in $X$, and $x=(x_1,x_2,\cdots,x_n)$ be a vector in $X$. Note that $a_j,x_j\in X_j$ for $j=1,2,\cdots,n$. @@ -517,12 +531,31 @@ $$ f(x)=\sum_{a\in X} f(a)\epsilon_a(x) $$ +
+Proof + +This basically follows the same rascal as the previous proof. This time, the epsilon function only returns $1$ when $x_j=a_j$ for all $j=1,2,\cdots,n$. + +$$ +f(x)=\sum_{a\in X} f(a)\epsilon_a(x)=f(x) +$$ + +QED. + +
+ +#### Definition of tensor product of basis elements + **The tensor product of basis elements** is defined by $$ -\epsilon_a=\epsilon_{a_1}^{(1)}\otimes \epsilon_{a_2}^{(2)}\otimes \cdots \otimes \epsilon_{a_n}^{(n)} +\epsilon_a\coloneqq\epsilon_{a_1}^{(1)}\otimes \epsilon_{a_2}^{(2)}\otimes \cdots \otimes \epsilon_{a_n}^{(n)} $$ +This is a basis of $\mathscr{H}$, here $\mathscr{H}$ is the set of all functions from $X=X_1\times X_2\times \cdots \times X_n$ to $\mathbb{C}$. + +#### Definition of tensor product of two finite-dimensional Hilbert spaces + **The tensor product of two finite-dimensional Hilbert spaces** (in $\mathscr{H}$) is defined by Let $\mathscr{H}_1$ and $\mathscr{H}_2$ be two finite dimensional Hilbert spaces. Let $u_1\in \mathscr{H}_1$ and $v_1\in \mathscr{H}_2$. @@ -531,7 +564,7 @@ $$ u_1\otimes v_1 $$ -is a bi-anti-linear map from $\mathscr{H}_1\otimes \mathscr{H}_2$ to $\mathbb{F}$ (in this case, $\mathbb{C}$). And $\forall u\in \mathscr{H}_1, v\in \mathscr{H}_2$, +is a bi-anti-linear map from $\mathscr{H}_1\times \mathscr{H}_2$ (the Cartesian product of $\mathscr{H}_1$ and $\mathscr{H}_2$, a tuple of two elements where first element is in $\mathscr{H}_1$ and second element is in $\mathscr{H}_2$) to $\mathbb{F}$ (in this case, $\mathbb{C}$). And $\forall u\in \mathscr{H}_1, v\in \mathscr{H}_2$, $$ (u_1\otimes v_1)(u, v)=\langle u,u_1\rangle \langle v,v_1\rangle @@ -540,7 +573,7 @@ $$ We call such forms **decomposable**. The tensor product of two finite-dimensional Hilbert spaces, denoted by $\mathscr{H}_1\otimes \mathscr{H}_2$, is the set of all linear combinations of decomposable forms. Represented by the following: $$ -(\sum_{i=1}^n a_i u_i\otimes v_i)(u, v)=\sum_{i=1}^n a_i \langle v,u_i\rangle \langle v_i,u\rangle +\left(\sum_{i=1}^n a_i u_i\otimes v_i\right)(u, v) \coloneqq \sum_{i=1}^n a_j(u_j\otimes v_j)(u,v)=\sum_{i=1}^n a_i \langle v,u_i\rangle \langle v_i,u\rangle $$ Note that $a_i\in \mathbb{C}$ for complex-vector spaces. @@ -550,7 +583,7 @@ This is a linear space of dimension $\dim \mathscr{H}_1\times \dim \mathscr{H}_2 We define the inner product of two elements of $\mathscr{H}_1\otimes \mathscr{H}_2$ ($u_1\otimes v_1:(\mathscr{H}_1\otimes \mathscr{H}_2)\to \mathbb{C}$, $u_2\otimes v_2:(\mathscr{H}_1\otimes \mathscr{H}_2)\to \mathbb{C}$ $\in \mathscr{H}_1\otimes \mathscr{H}_2$) by $$ -\langle u_1\otimes v_1, u_2\otimes v_2\rangle=\langle u_1,u_2\rangle \langle v_1,v_2\rangle=(u_1\otimes v_1)(u_2,v_2) +\langle u_1\otimes v_1, u_2\otimes v_2\rangle\coloneqq\langle u_1,u_2\rangle \langle v_1,v_2\rangle=(u_1\otimes v_1)(u_2,v_2) $$ ### Tensor products of linear operators diff --git a/content/contact.md b/content/contact.md index 0b5549d..7dab5a8 100644 --- a/content/contact.md +++ b/content/contact.md @@ -1,10 +1,12 @@ # Contact -This page is mainly maintained by [Trance-0](https://github.com/Trance-0). +This page is mainly maintained by [Zheyuan Wu](https://resume.trance-0.com). ## Contact info [GitHub](https://github.com/Trance-0) +[LinkedIn](https://www.linkedin.com/in/zheyuan-wu-742b1a227) + [Email](mailto:me@trance-0.com) diff --git a/favicon.ico b/public/favicon.ico similarity index 100% rename from favicon.ico rename to public/favicon.ico diff --git a/favicon.svg b/public/favicon.svg similarity index 100% rename from favicon.svg rename to public/favicon.svg