When Phase Is Measurable

Relative phase between components of a superposition is physically real and measurable; a global phase on the whole state is not.

Two kinds of phase

Write a qubit as |psi> = a|0> + b e^{i phi} |1>. The angle phi is a relative phase between the two amplitudes. Multiplying the entire state by e^{i theta} instead gives a global phase. These behave completely differently under measurement: relative phase changes physical predictions, global phase never does.

Why global phase is unobservable

Any measurement probability is Tr(P rho), and the density matrix of e^{i theta}|psi> equals that of |psi> because the phase cancels between bra and ket. No experiment can distinguish |psi> from e^{i theta}|psi>. Formally, physical states are rays in Hilbert space, not vectors, and the true state space is a projective space where global phase has been quotiented out.

Why relative phase is real

Relative phase is invisible in the computational basis: |a|^2 and |b|^2 do not depend on phi. But it becomes visible after a basis change that lets the components interfere. Apply a Hadamard to a|0> + b e^{i phi}|1> and the outcome probabilities depend on phi through a cos(phi) term. Ramsey interferometry exploits exactly this: put a qubit in superposition, let a phase accumulate, then interfere and read the phase off the fringe.

import numpy as np
def probs(phi):
    psi = np.array([1, np.exp(1j*phi)])/np.sqrt(2)
    H = np.array([[1,1],[1,-1]])/np.sqrt(2)
    return np.abs(H @ psi)**2
print(np.round(probs(0),3), np.round(probs(np.pi),3))

At phi = 0 the state collapses toward |0>; at phi = pi it collapses toward |1>. The same magnitudes, different relative phase, give opposite results after interference.

Practical consequences

Because relative phase carries information, controlled dephasing is a form of error, and phase gates are essential computational primitives. Because global phase is meaningless, it is standard to fix gate matrices only up to an overall phase, and hardware need not track it. Phase estimation algorithms are built to convert an unknown relative phase into a binary answer by controlled interference and readout.