Global vs Relative Phase

A precise account of why an overall phase factor is physically irrelevant while phase differences within a superposition are not.

The projective structure of state space

Quantum states that differ only by an overall factor e^{i theta} represent the same physical situation. The genuine state space is therefore not the Hilbert space itself but the set of rays, called projective Hilbert space. For a single qubit this is the Bloch sphere: the four real parameters of a two-component complex vector collapse to two angles once normalization and global phase are removed.

Density matrices erase global phase automatically

The density matrix rho = |psi> -> e^{i theta}|psi>, so representing states as density operators builds in the irrelevance of global phase from the start. Every observable prediction is a function of rho, which is why global phase can never appear in any probability or expectation value.

Relative phase survives in rho

For |psi> = a|0> + b|1>, the off-diagonal element of rho is a b*, whose argument is the relative phase. These coherences are what interference reads out. A process that drives the off-diagonals to zero while leaving the diagonal populations intact is pure dephasing; it destroys relative phase without changing which-state probabilities in the computational basis.

The diagonal entries are populations; the off-diagonal entries carry the relative phase. Global phase would multiply the whole matrix by e^{i theta} times its conjugate, i.e. by one, so it cannot appear.

Why the distinction matters in a circuit

A single-qubit Z gate and its global-phase-shifted cousin -Z act identically on any lone qubit, so implementations may ignore the difference. But once that qubit is a control, the phase becomes relative to the other branch and becomes physical: a controlled-Z is not a controlled-(-Z). This is why phases that look ignorable on isolated qubits must be tracked carefully inside controlled operations and multi-qubit gates.