update

pages/Math401/Math401_P1_1.md

@@ -0,0 +1,4 @@
+# Math 401, Paper 1, Side note 1: Quantum information theory and Measure concentration
+
+## MM space
+

pages/Math401/Math401_T5.md

@@ -34,7 +34,7 @@ The composition operator $\psi\mapsto U\psi=\psi\circ T$, where $T$ is a measure
 
 ### Spring-mass system
 
-![Spring-mass system](https://notenextra.com/Math401/Spring-mass_system.png)
+![Spring-mass system](https://notenextra.trance-0.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)
 
@@ -56,6 +56,148 @@ A linear operator $A$ on a Hilbert space $\mathscr{H}$ is said to be Hermitian i
 
 It is skew-Hermitian if $\langle A\psi,\phi\rangle=-\langle\psi,A\phi\rangle$.
 
+## Section 3: Hamiltonians and the Schrödinger equation (finite dimensional version)
 
+the problem of solving Schrödinger equation is at its core about studying the spectral theory of the Hamiltonian operator.
 
+### Dynamics in 2-dimensional (_2 level_) systems (qubit)
 
+In previous sections, we know that any self-adjoint matrix has the form $x_0+\vec{x}\cdot \sigma$, where $\sigma$ is the Pauli matrices.
+
+And $(x_0,\vec{x})\in\mathbb{R}^4$ is a point in $\mathbb{R}^4$.
+
+The general form (time-independent) of the Hamiltonian for a 2-level system is:
+
+$$
+H=\begin{pmatrix}
+x_0+x_3 & x_1-ix_2 \\
+x_1+ix_2 & -x_0+x_3
+\end{pmatrix}
+$$
+
+Parameterizing the curves in Bloch space generated by Hamiltonian. In physical dimension of $\vec{x}=\omega\hbar\vec{s}$, $\omega>0$. $\omega\hbar$ is the physical dimension of energy.
+
+we have:
+
+$$
+H=\omega\hbar\begin{pmatrix}
+s_3 & s_1-is_2 \\
+s_1+is_2 & -s_3
+\end{pmatrix}
+$$
+
+[Continue on the orbits of states in the Bloch sphere] skip for now.
+
+## Section 4: Transition probability, probability amplitudes and the Born rule
+
+the modulus squared of a probability amplitude is the probability of the corresponding state.
+
+### Basic definitions in transition probability
+
+#### Definition of probability amplitude
+
+For a n-dimensional Hilbert space $\mathscr{H}$, the system is initially in a pure state give by the unit vector $|\psi_0\rangle\in\mathscr{H}$, thus with the density operator $\rho_0=|\psi_0\rangle\langle\psi_0|$.
+
+Then the state at time $t_1$ is given by $|\psi_1\rangle=A|\psi_0\rangle$, where $A\in U(n)$ is a unitary operator.
+
+Then the density operator at time $t_1$ is given by $\rho_1=|\psi_1\rangle\langle\psi_1|=A|\psi_0\rangle\langle\psi_0|A^*=A\rho_0A^*$.
+
+The entry of $A$ are $a_{ij}=\langle i|A|j\rangle$. where $|i\rangle$ is the basis of $\mathscr{H}$.
+
+The $a_{ij}$ are the probability amplitudes of the transition from state $|i\rangle$ to state $|j\rangle$.
+
+#### Definition of transition probability
+
+Given above, the transition probability from state $|i\rangle$ to state $|j\rangle$ is given by:
+
+$$
+|a_{ij}|^2
+$$
+
+#### Sum over paths
+
+To each path of classical states, path $j\to i: i_0=j,i_1,i_2,\cdots,i_l=i$, we associates the probability amplitude of the path given by:
+
+$$
+|\text{path}(j\to i)\rangle=\langle i_0|i_1\rangle\langle i_1|i_2\rangle\cdots\langle i_{l-1}|i_l\rangle
+$$
+
+The probability of the path is given by:
+
+$$
+\operatorname{Prob}(i|j)=\left|\sum_{\text{all paths}j\to i \text{ with } l \text{ steps}}|\text{path}(j\to i)\rangle\right|^2
+$$
+
+### Measuring a qubit
+
+#### Definition of qubit
+
+A qubit is a 2-level quantum system.
+
+One example of qubit is the photon polarization.
+
+#### Measurement of a qubit
+
+The measurement of a qubit is a map fro the space of density operators, to a point on the intervals $[0,1]$.
+
+This gives a probability distribution on the interval $[0,1]$ in our classical probability space.
+
+![Measurement of a qubit](https://notenextra.trance-0.com/Math401/Measurement_of_a_qubit.png)
+
+Here $p=\cos^2(\theta)\in[0,1]$. is the probability of the state being in the state $|0\rangle$.
+
+The north pole on the Bloch sphere gives probability $1$ for the state being in the state $|0\rangle$.
+
+The south pole on the Bloch sphere gives probability $1$ for the state being in the state $|1\rangle$.
+
+The equator on the Bloch sphere gives probability $1/2$ for the state being in the state $|0\rangle$ or $|1\rangle$.
+
+### Projective measurement of an $N$-qubit system
+
+For $N$ qubits, the pure quantum state $\rho=|\psi\rangle\langle\psi|$ represented by the state vector $|\psi\rangle\in\mathscr{H}^{\otimes N}=\mathscr{H}\otimes\cdots\otimes\mathscr{H}(\mathscr{H}=\mathbb{C}^2)$.
+
+This produces as output the random variable $X\in \{0,1\}^N$. $X=(a_1,a_2,\cdots,a_N)$, where $a_i\in \{0,1\}$.
+
+By the Born rule,
+
+$$
+\operatorname{Prob}(X=(a_1,a_2,\cdots,a_N))=\left|\langle a_1a_2\cdots a_N|\psi\rangle\right|^2
+$$
+
+where $\langle a_1a_2\cdots a_N|\psi\rangle=\langle a_1|\otimes\langle a_2|\otimes\cdots\otimes\langle a_N|\psi\rangle$.
+
+The input vector state $|\psi\rangle$ is a unit vector in $\mathscr{H}^{\otimes N}$.
+
+This can be written as a tensor product of the basis vectors:
+
+$$
+|\psi\rangle=\sum_{a_1,a_2,\cdots,a_N} c_{a_1,a_2,\cdots,a_N}|a_1a_2\cdots a_N\rangle
+$$
+
+where $c_{a_1,a_2,\cdots,a_N}\in\mathbb{C}$.
+
+The probability distribution of the post-measurement **classical random variable** $X$ can be represented as a point in the $2^N-1$ dimensional simplex of all probability distributions on the set $\{0,1\}^N$.
+
+$$
+\mathscr{P}(\{0,1\}^N)=\left\{(p_1,p_2,\cdots,p_{2^N})\in\mathbb{R}^{2^N}:p_i\geq 0,\sum_{i=1}^{2^N}p_i=1\right\}
+$$
+
+![Simplex of all probability distributions on the set $\{0,1\}^N$](https://notenextra.trance-0.com/Math401/Simplex_of_all_probability_distributions_on_the_set_01N.png)
+
+here we use the binary representation for the index $i$ in the diagram.
+
+#### Pure versus mixed states
+
+A pure state is a state that is represented by a unit vector in $\mathscr{H}^{\otimes N}$.
+
+A mixed state is a state that is represented by a density operator in $\mathscr{H}^{\otimes N}$. (convex combination of pure states)
+
+if $\rho_j=|\psi_j\rangle\langle\psi_j|$, then $\rho=\sum_{j=1}^N p_j\rho_j$ is a mixed state, where $p_j\geq 0$ and $\sum_{j=1}^N p_j=1$.
+
+#### Projective measurement of subsystem and partial trace
+
+This section is related to quantum random walk and we will skip it for now.
+
+## Section 5: Quantum random walk
+
+This part is skipped, it is an interesting topic, but it is not the focus of my research for now.

pages/Math401/Math401_T6.md

@@ -1 +1,2 @@
-# Math 401, Topic 6: Postulates of quantum theory and measurement operations
+# Math 401, Topic 6: Postulates of quantum theory and measurement operations
+

pages/Math401/_meta.js

@@ -20,4 +20,5 @@ export default {
         type: 'separator'
     },
     Math401_P1: "Math 401, Paper 1: Concentration of measure effects in quantum information (Patrick Hayden)",
+    Math401_P1_1: "Math 401, Paper 1, Side note 1: Quantum information theory and Measure concentration",
 }