For questions related to physical realizations of quantum bits i.e. quantum systems living in a 2-dimensional Hilbert space.
A qubit (quantum binary digit) is the quantum analogue of the classical bit.
As such, it is a two-level system and some example physical systems that can be used as a qubit include:
- the ground and first excited state of an atom, or other equivalent system
- polarisation of a photon
- two spatial or temporal modes of a photon
- spin of an electron/atom/quantum dot
- charge or (direction of) current of a system
- squeezed light
Mathematically, a qubit lives on a 2-dimensional Hilbert space. The 'computational basis' is denoted by the states $$\left|0\right> = \begin{pmatrix}1 \\\ 0 \end{pmatrix} \quad \text{and}\quad\left|1\right>=\begin{pmatrix}0 \\\ 1\end{pmatrix},$$
so that any single qubit state can be written as $\left|\psi\right> = \alpha\left|0\right>+\beta\left|1\right>$ (known as superposition), with the constraint that $\lvert\alpha\rvert^2+\lvert\beta\rvert^2=1$. Taking out a global phase allows these to be written as $\alpha=\cos\frac{\theta}{2}$ and $\beta=e^{i\phi}\sin\frac{\theta}{2}$.
Entangling $n$ qubits allows for creation of larger states living in a $2^n$-dimensional Hilbert space. For more information, see also the quantum-entanglement tag.
Higher dimensional versions of the qubit, known as the qutrit and qudit also exist, although are less widely used.
