A quantum state is a complete mathematical description of a quantum system. For a single qubit, a quantum state is a unit vector in a two-dimensional complex Hilbert space; for $n$ qubits, the state lives in a $2^n$-dimensional space formed by the tensor product of the individual qubit spaces.

Pure states are written as ket vectors $\lvert\psi\rangle = \alpha\lvert 0\rangle + \beta\lvert 1\rangle$, where $\alpha, \beta \in \CC$ and $|\alpha|^2 + |\beta|^2 = 1$. The squared magnitudes $|\alpha|^2$ and $|\beta|^2$ give the probabilities of measuring $\lvert 0\rangle$ and $\lvert 1\rangle$ respectively. When a system cannot be described by a single ket — because it is entangled with the environment or prepared as a statistical mixture — the state is instead represented by a density matrix $\rho$.

The set of single-qubit pure states maps one-to-one onto the surface of the Bloch sphere, where the north and south poles are $\lvert 0\rangle$ and $\lvert 1\rangle$, the equatorial $x$-axis states are $\lvert +\rangle$ and $\lvert -\rangle$, and the equatorial $y$-axis states are $\lvert +i\rangle$ and $\lvert -i\rangle$. Multi-qubit states can be entangled, meaning they cannot be written as a product of individual qubit states; the Bell states are the canonical two-qubit entangled states.