CSE 599Q: (Our) Principles of Quantum Mechanics

A summary of our informal set of quantum mechanical principles:

1. (Superposition) If a quantum system can be in one of $d$ basic states $\ket{1},\ket{2},\ldots,\ket{d}$, then it can also be in any superposition $\alpha_1 \ket{1} + \alpha_2 \ket{2} + \cdots + \alpha_d \ket{d}$ where $\alpha_1,\alpha_2,\ldots,\alpha_d,$ are complex numbers and $|\alpha_1|^2 + |\alpha_2|^2 + \cdots + |\alpha_d|^2 = 1$.

This makes the set of quantum states of (discrete) quantum system into a finite-dimensional complex vector space. A $d$-dimensional quantum state is therefore a unit vector $\ket{\psi} \in \C^d$.

2. (Measurement) If $\ket{v_1},\ldots,\ket{v_d}$ is an orthornormal basis of $\C^d$ and

$\ket{\psi} = \beta_1 \ket{v_1} + \cdots + \beta_d \ket{v_d}$

is a quantum state, then measuring $\ket{\psi}$ in this basis yields outcome “$\ket{v_j}$” with probability $\abs{\beta_j}^2$, and in this case $\ket{\psi}$ collapses to the state $\ket{v_j}$.

(Partial measurements) Suppose that $\ket{\psi} \in \C^{m} \otimes \C^n$ is a quantum state. Recall that we can write

$\ket{\psi} = \sum_{j=1}^m \sum_{k=1}^n \alpha_{jk} \ket{j_A}\ket{k_B},$

where $\{\ket{1_A},\ldots,\ket{m_A}\}$ is an orthornormal basis for $\C^m$, and $\{\ket{1_B},\ldots,\ket{n_B}\}$ is an orthornormal basis for $\C^n$.

If we perform a measurement on the first qubit in the $\{\ket{1_A},\ldots,\ket{m_A}\}$ basis, then the measured outcome is “$\ket{j_A}$” with probability

$\sum_{k=1}^n |\alpha_{jk}|^2\,,$

and in this case the state collapses to

$\frac{1}{\left(\sum_{k=1}^n |\alpha_{jk}|^2\right)^{1/2}} \sum_{k=1}^n \alpha_{jk} \ket{j_A} \ket{k_B}$
3. (Unitary evolution) If $U$ is a $d \times d$ unitary matrix and $\ket{\psi} \in \mathbb{C}^d$ is a quantum state, then one can “physically” realize the new state $U \ket{\psi}$.

4. (Composing quantum systems) If $\ket{\psi} \in \C^d$ and $\ket{\phi} \in \C^{d'}$ are independent quantum states, then their joint state is given by $\ket{\psi} \otimes \ket{\phi} \in \C^{d \times d'}$.

5. (Evolution of subsystems) Suppose that $\ket{\psi} \in \C^{m} \otimes \C^n$ is a quantum state. Recall that we can write $\ket{\psi} = \sum_{j=1}^m \sum_{k=1}^n \alpha_{jk} \ket{j_A}\ket{k_B},$ since $\{\ket{1_A},\ldots,\ket{m_A}\}$ is an orthornormal basis for $\C^m$, and $\{\ket{1_B},\ldots,\ket{n_B}\}$ is an orthornormal basis for $\C^n$. And $\ket{a}\ket{b}$ is shorthand for the tensor product $\ket{a} \times \ket{b}$.

Suppose now that $U \in \C^{m \times m}$ is a unitary matrix operating only on the first subsystem. Then the evolution of $\ket{\psi}$ is given by

$(U \otimes I) \ket{\psi} = \sum_{j=1}^m \sum_{k=1}^n \alpha_{jk} (U \ket{j_A}) \otimes \ket{k_B},$

where $I$ is the $n \times n$ identity matrix.