Global vs Relative Phase
Global phase is physically unobservable, but relative phase between amplitudes is real and drives quantum interference.
Two kinds of phase
Quantum amplitudes are complex, so they carry phase. There are two distinct notions. A global phase multiplies the entire state by e^{i alpha}. A relative phase is the phase difference between the amplitudes of different basis states. Only the second is physical.
Global phase is invisible
The states |psi> and e^{i alpha}|psi> give identical measurement statistics in every basis, because the Born rule uses |amplitude|^2 and the global factor cancels. No experiment can distinguish them. This is why physical states are properly described as rays rather than vectors — the overall phase is a gauge freedom.
Relative phase is everything
Compare (|0>+|1>)/sqrt(2) with (|0>-|1>)/sqrt(2). In the computational basis both give 50/50, so relative phase seems hidden. But apply a Hadamard: the first returns |0>, the second returns |1>. The relative phase, invisible in one basis, becomes a definite, distinguishable outcome in another. This convertibility is the engine of interference.
- Global: e^{i alpha}(a|0>+b|1>) — same physics, ignore it
- Relative: a|0> + e^{i phi} b|1> — real, measurable via basis change
On the Bloch sphere
Global phase does not move the point on the Bloch sphere at all, confirming its irrelevance. Relative phase phi is the longitude, so phase gates spin the state around the vertical axis. Two states differing only by relative phase are genuinely different points.
Why algorithms care
Quantum algorithms encode partial results into relative phases and then use interference to turn those phases into measurable amplitudes. Phase kickback, the phase estimation routine, and Grover's search all manipulate relative phase. Ignoring global phase, in contrast, simplifies bookkeeping without losing any physics.