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.

• $\lvert 0 \rangle$ (Zero state) — computational basis state, $+z$ Bloch pole (north pole)
• $\lvert 1\rangle$ (One state) — computational basis state, $-z$ Bloch pole (south pole)
• $\lvert +\rangle$ (Plus state) — equal superposition with positive amplitudes, $+x$ Bloch pole
• $\lvert -\rangle$ (Minus state) — equal superposition with opposite signs, $-x$ Bloch pole
• $\lvert +i\rangle$ (Plus-i state) — equal superposition with $+i$ relative phase, $+y$ Bloch pole
• $\lvert -i\rangle$ (Minus-i state) — equal superposition with $-i$ relative phase, $-y$ Bloch pole
• Bell states — four maximally entangled two-qubit states
  • $\lvert\Phi^+\rangle$ (Phi-plus)
  • $\lvert\Phi^-\rangle$ (Phi-minus)
  • $\lvert\Psi^+\rangle$ (Psi-plus)
  • $\lvert\Psi^-\rangle$ (Psi-minus)
• W state — three-qubit entangled state, robust under qubit loss
• GHZ state — three-or-more-qubit entangled state, fragile generalisation of the Bell state