Representations of Spin-1/2 Systems

To describe the states of spin-$½$ qubits, we can use the three representations introduced for other quantum states: abstract quantum state vectors (right vectors), column vectors, and graphical representations in state space. The state vectors are often written as $\ket{\uparrow}$ for spin-up and $\ket{\downarrow}$ for spin-down. Those are usually chosen to be the basis states for the system. A spin-$½$ state space diagram is shown in the following figure: