The Importance of Phase in Quantum Mechanics

Although when finally measuring, the phase does not play an impact on the probability of measuring a state, it does play an impact in the quantum circuit before measurement when quantum gates are applied. While an entangled state is going through a quantum circuit, its phase is affected by its interactions with the gates and can lead to different probabilities. For example, $|+\rangle$ and $|-\rangle$ when measured have the same 50-50 probability of measuring $|0\rangle$ or $|1\rangle$ (the phase does not impact measurement), but in $H|+\rangle → |0\rangle$ and $H|-\rangle → |1\rangle$, the phase does impact the final measurement.

The phase stores extra information, which can be extracted through gates like the Hadamard gate.