Relative Phases in Quantum States

We showed earlier that the relative sign between the amplitudes in a quantum superposition state leads to observable results, that is, to different measurement outcomes. In particular, if the superposition state is prepared in one set of basis states, but measured with respect to a different set of basis states, the measurement outcomes can tell us the relative sign between the superposition amplitudes in the original basis states. We will now revisit this using complex exponentials.

Let’s start with a general phase difference between the amplitudes in superposition states. We start with a superposition state for a single qubit, and for the sake of simplicity we assume the amplitudes have a magnitude $1/ \sqrt{2}:$

$$|s⟩ = \frac{1}{\sqrt{2}} \left[|0⟩ + e^{i\phi} |1⟩\right].$$