diff --git a/pages/Math401/Math401_P1.md b/pages/Math401/Math401_P1.md new file mode 100644 index 0000000..242fa34 --- /dev/null +++ b/pages/Math401/Math401_P1.md @@ -0,0 +1,34 @@ +# Math 401, Paper 1: Concentration of measure effects in quantum information (Patrick Hayden) + +[PDF](https://www.ams.org/books/psapm/068/2762144) + +## Quantum codes + +### Preliminaries + +#### Daniel Gottesman's mathematics of quantum error correction + +##### Quantum channels + +Encoding channel and decoding channel + +#### Quantum capacity for a quantum channel + +#### Lloyd-Shor-Devetak theorem + +### Surprise in high-dimensional quantum systems + +#### Levy's lemma + +### Random states and random subspaces + +#### ebits and qbits + +### Superdense coding of quantum states + +### Consequences for mixed state entanglement measures + +#### Quantum mutual information + +### Multipartite entanglement + diff --git a/pages/Math401/Math401_T5.md b/pages/Math401/Math401_T5.md index cff8b92..d6f84a3 100644 --- a/pages/Math401/Math401_T5.md +++ b/pages/Math401/Math401_T5.md @@ -12,19 +12,15 @@ The theory of dynamics is the study of properties of orbits. #### Definition of measure-preserving map -Let $P$ be a probability measure on a $\sigma$-algebra $\mathscr{F}$ of subsets of $\Omega$. A measurable transformation $T:\Omega\to\Omega$ is said to be measure-preserving if for all random variables $\psi:\Omega\to\mathbb{R}$, we have $\mathbb{E}(\psi\circ T)=\mathbb{E}(\psi)$, that is: +Let $P$ be a probability measure on a $\sigma$-algebra $\mathscr{F}$ of subsets of $\Omega$. (that is, $P:\mathscr{F}\to$ anything) A measurable transformation $T:\Omega\to\Omega$ is said to be measure-preserving if for all random variables $\psi:\Omega\to\mathbb{R}$, we have $\mathbb{E}(\psi\circ T)=\mathbb{E}(\psi)$, that is: $$ \int_\Omega (\psi\circ T)(\omega)dP(\omega)=\int_\Omega \psi(\omega)dP(\omega) $$ -#### Definition of ergodic map +Example: -A measurable transformation $T:\Omega\to\Omega$ is said to be ergodic if for all random variables $\psi:\Omega\to\mathbb{R}$, we have $\mathbb{E}(\psi\circ T)=\mathbb{E}(\psi)$, that is: - -$$ -\int_\Omega (\psi\circ T)(\omega)dP(\omega)=\int_\Omega \psi(\omega)dP(\omega) -$$ +The doubling map $T:\Omega\to\Omega$ is defined as $T(x)=2x\mod 1$, is a Lebesgue measure preserving map on $\Omega=[0,1]$. #### Definition of isometry @@ -34,4 +30,32 @@ The composition operator $\psi\mapsto U\psi=\psi\circ T$, where $T$ is a measure The composition operator $\psi\mapsto U\psi=\psi\circ T$, where $T$ is a measure preserving map defined on $\mathscr{H}=L^2(\Omega,\mathscr{F},P)$ is unitary of $\mathscr{H}$ if $U$ is an isometry and $T$ is invertible with measurable inverse. -## Section 2: Continuous time (classical) dynamical systems \ No newline at end of file +## Section 2: Continuous time (classical) dynamical systems + +### Spring-mass system + +![Spring-mass system](https://notenextra.com/Math401/Spring-mass_system.png) + +The pure state of the system is given by the position and velocity of the mass. $(x,v)$ is a point in $\mathbb{R}^2$. $\mathbb{R}^2$ is the state space of the system. (or phase space) + +The motion of the system in its state space is a closed curve. + +$$ +\Phi_t(x,v)=\left(\cos(\omega t)x-\frac{1}{\omega}\sin(\omega t)v, \cos(\omega t)v-\omega\sin(\omega t)x\right) +$$ + +Such system with closed curve is called **integrable system**. Where the doubling map produces orbits having distinct dynamical properties (**chaotic system**). + +> Note, some section is intentionally ignored here. They are about in the setting of operators on Hilbert spaces, the evolution of (classical, non-dissipative e.g. linear spring-mass system) system, is implemented by a one-parameter group of unitary operators. +> +> The detailed construction is omitted here. + +#### Definition of Hermitian operator + +A linear operator $A$ on a Hilbert space $\mathscr{H}$ is said to be Hermitian if $\forall \psi,\phi\in$ **domain of $A$**, we have $\langle A\psi,\phi\rangle=\langle\psi,A\phi\rangle$. + +It is skew-Hermitian if $\langle A\psi,\phi\rangle=-\langle\psi,A\phi\rangle$. + + + + diff --git a/pages/Math401/Math401_T6.md b/pages/Math401/Math401_T6.md new file mode 100644 index 0000000..7f412a2 --- /dev/null +++ b/pages/Math401/Math401_T6.md @@ -0,0 +1 @@ +# Math 401, Topic 6: Postulates of quantum theory and measurement operations \ No newline at end of file diff --git a/pages/Math401/Math401_T7.md b/pages/Math401/Math401_T7.md new file mode 100644 index 0000000..055e35e --- /dev/null +++ b/pages/Math401/Math401_T7.md @@ -0,0 +1 @@ +# Math 401, Topic 7: Basic of quantum circuits \ No newline at end of file diff --git a/pages/Math401/_meta.js b/pages/Math401/_meta.js index 24b47be..0c83d91 100644 --- a/pages/Math401/_meta.js +++ b/pages/Math401/_meta.js @@ -6,9 +6,18 @@ export default { Math401_N1: "Math 401, Notes 1", Math401_N2: "Math 401, Notes 2", Math401_N3: "Math 401, Notes 3", + "---":{ + type: 'separator' + }, Math401_T1: "Math 401, Topic 1: Probability under language of measure theory", Math401_T2: "Math 401, Topic 2: Finite-dimensional Hilbert spaces", Math401_T3: "Math 401, Topic 3: Separable Hilbert spaces", Math401_T4: "Math 401, Topic 4: The quantum version of probabilistic concepts", Math401_T5: "Math 401, Topic 5: Introducing dynamics: classical and non-commutative", + Math401_T6: "Math 401, Topic 6: Postulates of quantum theory and measurement operations", + Math401_T7: "Math 401, Topic 7: Basic of quantum circuits", + "---":{ + type: 'separator' + }, + Math401_P1: "Math 401, Paper 1: Concentration of measure effects in quantum information (Patrick Hayden)", } \ No newline at end of file diff --git a/public/Math401/Spring-mass_system.png b/public/Math401/Spring-mass_system.png new file mode 100644 index 0000000..beab1e4 Binary files /dev/null and b/public/Math401/Spring-mass_system.png differ