Why Global Phase Is Unobservable

Multiplying a whole state by a phase changes no measurement outcome, so physical states are rays and global phase is discarded.

A gauge freedom

Multiplying an entire quantum state by a phase factor e^{i alpha} produces a state that is physically indistinguishable from the original. This global phase is a gauge freedom: it can be chosen arbitrarily without affecting any prediction. Understanding why sharpens intuition about what is and is not physical in a state vector.

The proof from the Born rule

Every measurable prediction comes from the Born rule, which uses squared magnitudes of amplitudes. For a state e^{i alpha}|psi>, any outcome probability is ||^2 = |e^{i alpha}|^2 ||^2 = ||^2, since |e^{i alpha}| = 1. The phase cancels in every basis, so no experiment can detect it.

States are rays

Because |psi> and e^{i alpha}|psi> are the same physically, the true space of states is the set of rays — equivalence classes of vectors differing by a global phase — rather than vectors themselves. This is called projective Hilbert space. The Bloch sphere is exactly this ray space for one qubit: global phase does not move the point.

Relative phase survives

The contrast is essential. A phase applied to only part of a superposition — the relative phase between amplitudes — is physical and drives interference. In a|0> + e^{i phi} b|1>, the phi is measurable by changing basis; only a common factor multiplying both terms is invisible.

Practical consequences

Two gates that differ only by an overall phase are physically equivalent, which simplifies gate synthesis and lets compilers ignore global phase. But care is required with controlled versions: a global phase on a target becomes a relative phase when that gate is controlled, so a phase that was ignorable can become observable through phase kickback. Knowing when phase is global and when it is not prevents subtle circuit errors.