Global and Relative Phases

In quantum mechanics, a global phase is a complex number of magnitude 1 (i.e., $e^{i\theta}$ for some real $\theta$) that multiplies an entire quantum state.

The key principle is that two states differing only by a global phase are physically indistinguishable—they give identical measurement probabilities for any measurement.

Consider a quantum state $|\psi\rangle$ and the state $\ket{\phi} =e^{i\theta}|\psi\rangle$.

For any measurement basis, these states yield the same probabilities. To see why, recall that the probability of measuring outcome $|k\rangle$ is given by Born's rule:

$$P_{\psi}(k) = |\langle k|\psi\rangle|^2$$

For the globally phase-shifted state:

$$P_{\phi}(k) = |\langle k|e^{i\theta}|\psi\rangle|^2 = |e^{i\theta}\langle k|\psi\rangle|^2 = |e^{i\theta}|^2|\langle k|\psi\rangle|^2 = 1 \cdot |\langle k|\psi\rangle|^2 = P_{\psi}(k) ...$$