Qubit

Remarks

Example: $|{+}\rangle = (|0\rangle + |1\rangle)/\sqrt{2}$ is a pure state, since $\abs{1/\sqrt2}^2 + \abs{1/\sqrt2}^2 = 1$. It is a superposition of the two basis states, and a measurement in the computational basis gives each outcome with probability $\tfrac12$.

Non-example: the vector $|0\rangle + |1\rangle$ has squared norm $2$, so it is not a unit vector and does not define a state as written.

For example, $|\psi\rangle$ and $-|\psi\rangle$ describe the same state. Since unit vectors that differ by a phase give the same state, the pure states form the complex projective line $\mathbb{CP}^1$. A system of $n$ qubits has Hilbert space $(\C^2)^{\otimes n}$, with computational basis $|x_1 \cdots x_n\rangle$ for $x_i \in \set{0, 1}$.