A summary of our informal set of quantum mechanical principles:
(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 \(\ket{\psi} \in \C^d\).
(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
\[\frac{1}{\left(\sum_{k=1}^n |\alpha_{jk}|^2\right)^{1/2}} \sum_{k=1}^n \alpha_{jk} \ket{j_A} \ket{k_B}\](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 \(U \ket{\psi}\).
(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 \(\ket{\psi} \otimes \ket{\phi} \in \C^{d \times d'}\).
(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.