A summary of our informal set of quantum mechanical principles:

1. (Superposition) If a quantum system can be in two basic states \(\ket{0}\) and \(\ket{1}\), then it can also be in a superposition \(\alpha \ket{0} + \beta \ket{1}\) where \(\alpha,\beta\) are complex numbers and \(\abs{\alpha}^2 + \abs{\beta}^2 = 1\).

   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 \(\ket{v_j}\).

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.

6. (Marginalizing quantum systems) For this, we need the notion of density matrices and the partial trace (these will come nearer the end of the course).