From fc1d1b0d8b49462540dbace1a645baead7eaffc4 Mon Sep 17 00:00:00 2001 From: Zheyuan Wu <60459821+Trance-0@users.noreply.github.com> Date: Thu, 26 Jun 2025 17:10:08 -0500 Subject: [PATCH] update --- pages/Math401/Math401_P1_1.md | 4 + pages/Math401/Math401_T5.md | 144 +++++++++++++++++- pages/Math401/Math401_T6.md | 3 +- pages/Math401/_meta.js | 1 + public/Math401/Measurement_of_a_qubit.png | Bin 0 -> 41065 bytes ...obability_distributions_on_the_set_01N.png | Bin 0 -> 28537 bytes 6 files changed, 150 insertions(+), 2 deletions(-) create mode 100644 pages/Math401/Math401_P1_1.md create mode 100644 public/Math401/Measurement_of_a_qubit.png create mode 100644 public/Math401/Simplex_of_all_probability_distributions_on_the_set_01N.png diff --git a/pages/Math401/Math401_P1_1.md b/pages/Math401/Math401_P1_1.md new file mode 100644 index 0000000..6a4c103 --- /dev/null +++ b/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 + diff --git a/pages/Math401/Math401_T5.md b/pages/Math401/Math401_T5.md index d6f84a3..bc4b7eb 100644 --- a/pages/Math401/Math401_T5.md +++ b/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. \ No newline at end of file diff --git a/pages/Math401/Math401_T6.md b/pages/Math401/Math401_T6.md index 7f412a2..3203af8 100644 --- a/pages/Math401/Math401_T6.md +++ b/pages/Math401/Math401_T6.md @@ -1 +1,2 @@ -# Math 401, Topic 6: Postulates of quantum theory and measurement operations \ No newline at end of file +# Math 401, Topic 6: Postulates of quantum theory and measurement operations + diff --git a/pages/Math401/_meta.js b/pages/Math401/_meta.js index 0c83d91..c64f3b0 100644 --- a/pages/Math401/_meta.js +++ 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