Postulate 1: Qubits and Superposition

There are competing opinions on how many postulates are needed to "lay down the law" of quantum theory. However, the first postulate of quantum theory remains pretty much unchanged in all the different formulations. It is about how to describe the states (think: physical configurations) of quantum systems.

We sometimes refer to the vectorspace that the quantum states live in as the state space.

Postulate 1: The state of individual quantum systems are described by unit vectors living in a (complete) complex vector space with an inner product, i.e., a Hilbert space. This Hilbert space is the state space of the system.

Two slightly technical notes on this postulate:

1. We will not concern ourselves with what being "complete" means here. It is a technical requirement for a Hilbert space that is unnecessarily technical for our purposes of learning quantum theory.
2. There are actually more general ways to write down the quantum state of a quantum system. For instance, there are what's known as mixed states. Unit vectors in a Hilbert Space refer only to a specific type of quantum state, known as a pure states. However, when first learning quantum theory, we can safely just concern ourselves with pure states, and move on to more general quantum states such as mixed states when one begins to develop the density matrix formulation of quantum theory.

The state space of an electron

For example, consider an electron that can be in one of two states: its ground state (or lowest energy state, close to the atom) and its excited state (or higher energy state, further away from the atom). 

We can model this with the following vectors:

$$|g\rangle = \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \quad |e\rangle = \begin{bmatrix} 0 \\ 1 \end{bmatrix}.$$

Where $g$ stands for the ground state, and $e$ stands for the excited state. (Recall we can put anything we want inside the kets for variables, and we often do! It often just reminds us of the physical context we're working in).

Computational Basis

Borrowing from the theory of classical computation, which concerns itself with bits ($0$ and $1$), we might also denote these states as:

$$|0\rangle = \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \quad |1\rangle = \begin{bmatrix} 0 \\ 1 \end{bmatrix},$$

which we call the computational basis. Note that basis vectors are always column vectors.

Notice these are two-dimensional vectors because, in this case, our quantum system only has two distinct, discrete values it can take (i.e. that we can measure). The ground state and the excited state.

Thus, for this physical system, $\mathbb{C}^2$ will suffice as the Hilbert space (state space) mentioned in Postulate 1.

...