Global phase vs relative phase

Amplitudes carry phase, and the last lesson put phase on the map as the angle around the equator. But not all phase is equal. One kind of phase is a bookkeeping artifact that no experiment can ever detect, and the other is the single most important resource in the whole subject - the thing every quantum algorithm spends to get its speedup. This lesson is the sharp line between the two, and why measurement sees one and is blind to the other.

Play first: two phase knobs, one that does nothing

The widget starts at $|+⟩$ and gives you both knobs. Sweep the global-phase slider and hunt for anything that moves: the Bloch vector, either set of measurement bars, any probability. Then sweep the relative-phase slider and watch the X-basis outcomes swing all the way from certain $|+⟩$ to certain $|-⟩$.

Global phase moved nothing you could measure - the arrows on the raw amplitudes spun, but every bar height and the Bloch point stayed put. Relative phase moved the Bloch vector around the equator and drove the X-basis outcome from one certainty to the opposite one, while the Z-basis stayed a flat 50/50. That second fact is the important one: relative phase is invisible in one basis and decisive in another.

Why global phase is invisible

This is not a convention, it falls straight out of the Born rule. Every probability you can ever observe is a squared magnitude of some amplitude. Multiply the whole state by $e^{i\gamma }$ and each amplitude picks up that factor, so each probability becomes:

The $e^{i\gamma }$ has magnitude 1, so squaring wipes it out every single time, in every basis. There is no measurement, no clever choice of experiment, that can reveal $\gamma$. Two states that differ only by a global phase are genuinely the same physical state - which is exactly why the Bloch sphere, having thrown global phase away, could still hold every distinct qubit.

Why relative phase is the whole game

Relative phase survives because it does not factor out. Take the equal superposition with a phase on the $|1⟩$ part and measure it in the $|+⟩,|-⟩$ basis. The amplitude landing on $|+⟩$ is a sum, and the phase sits inside that sum where squaring cannot remove it:

At $\varphi =0$ that is ${\cos }^{2}0=1$, a certain $|+⟩$; at $\varphi =\pi$ it is ${\cos }^2(\pi /2)=0$, a certain $|-⟩$. The two amplitudes interfered - reinforcing at $\varphi =0$, cancelling at $\varphi =\pi$ - and the relative phase is the dial that chose which. This is relative phase at work, and orchestrating exactly this cancel-or-reinforce across many amplitudes is what the real source of quantum speedup is made of.

import numpy as np

def probs(state):
    z = np.abs(state)**2                       # Z-basis: P(0), P(1)
    plus = np.array([1, 1]) / np.sqrt(2)
    minus = np.array([1, -1]) / np.sqrt(2)
    x = [abs(plus @ state)**2, abs(minus @ state)**2]   # X-basis: P(+), P(-)
    return z, x

psi = np.array([1, 1]) / np.sqrt(2)            # |+>

# Global phase: multiply the WHOLE state. Nothing observable moves.
probs(np.exp(1j * 1.2) * psi)                  # same z, same x as psi

# Relative phase pi on |1> only  ->  |->. Z basis unchanged, X basis flips.
psi_rel = np.array([1, -1]) / np.sqrt(2)
probs(psi_rel)                                 # z = [0.5, 0.5], x = [0.0, 1.0]

Check yourself

The global-versus-relative phase distinction follows Quantum Country^[1] and the state-space conventions of Nielsen & Chuang section 2.2^[2].