Global and Relative Phase

Two spinning coins can be perfectly synchronised or perfectly out of step, and if you only ever look at one of them you can never tell the difference — but line them up side by side and the mismatch leaps out. Quantum amplitudes carry exactly this kind of hidden timing, called phase. Some of it is a fiction of bookkeeping that nature refuses to reveal; the rest is as real and measurable as anything in physics. Telling the two apart is one of the most important habits of mind in quantum computing — and it is the whole reason quantum algorithms can do what they do.

A qubit |\psi\rangle = \alpha|0\rangle + \beta|1\rangle has complex amplitudes, and a complex number carries both a magnitude and a phase (an angle). There are two very different things phase can do here, and the entire lesson is the difference between them:

a global phase multiplies the whole state at once — e^{i\gamma}|\psi\rangle;
a relative phase sits between the two amplitudes — |0\rangle + e^{i\varphi}|1\rangle.

The first is invisible. The second is not. Keeping them straight is everything.

Global phase is invisible

Take any qubit and multiply the entire state by a phase factor e^{i\gamma} (a complex number of magnitude 1):

e^{i\gamma}|\psi\rangle = e^{i\gamma}\alpha\,|0\rangle + e^{i\gamma}\beta\,|1\rangle.

Does any experiment notice? A measurement only ever reports probabilities, and by the Born rule those are squared magnitudes. But |e^{i\gamma}| = 1, so the new amplitude of |0\rangle has probability

|e^{i\gamma}\alpha|^2 = |e^{i\gamma}|^2\,|\alpha|^2 = |\alpha|^2,

and likewise |e^{i\gamma}\beta|^2 = |\beta|^2. Every probability is untouched — not just in the computational basis, but in every basis, because the phase factors out before you ever square. No measurement, now or ever, can detect it. So we make it a rule:

e^{i\gamma}|\psi\rangle \;\equiv\; |\psi\rangle \quad\text{(same physical state)}.

The two are not merely similar — they are the same physical state wearing a different label. This is precisely the parameter the Bloch sphere is allowed to throw away: two complex amplitudes are four real numbers, normalisation removes one, and the unobservable global phase removes another, leaving just two angles — a point on a sphere.

Worked example: the phase cancels

Compare |1\rangle with -|1\rangle. The minus sign is a global phase, e^{i\pi} = -1, applied to the whole (one-term) state. Its measurement statistics: \Pr(1) = |-1|^2 = 1 either way. Identical.

A richer case: is \tfrac{1}{\sqrt2}\big(|0\rangle + |1\rangle\big) the same state as \tfrac{i}{\sqrt2}\big(|0\rangle + |1\rangle\big)? Factor the second: it is e^{i\pi/2}\cdot\tfrac{1}{\sqrt2}(|0\rangle + |1\rangle) — the same state times a global i. So yes, they are physically identical. The probabilities are \big|\tfrac{i}{\sqrt2}\big|^2 = \tfrac12 for each outcome, exactly as for |{+}\rangle. Whenever the same phase factor sits in front of both amplitudes, pull it out and discard it.

Relative phase is real

Now put the phase on only one amplitude. The general equal superposition is

|\psi_\varphi\rangle = \tfrac{1}{\sqrt2}\big(|0\rangle + e^{i\varphi}|1\rangle\big),

and the angle \varphi is the relative phase — the phase of |1\rangle's amplitude relative to |0\rangle's. You cannot factor \varphi out of both terms, so it does not cancel, and different \varphi genuinely give different states. The two most famous live at \varphi = 0 and \varphi = \pi:

|{+}\rangle = \tfrac{1}{\sqrt2}\big(|0\rangle + |1\rangle\big), \qquad |{-}\rangle = \tfrac{1}{\sqrt2}\big(|0\rangle - |1\rangle\big).

These differ by a single relative sign — a relative phase of \pi — yet they are orthogonal: their inner product is \langle +|-\rangle = \tfrac12(1)(1) + \tfrac12(1)(-1) = 0. Orthogonal states are as different as quantum states can be — perfectly, reliably distinguishable. A relative phase is not a bookkeeping ghost; it is a real, physical dial that moves a state around.

A dial you can point to

Picture the relative phase \varphi as an angle on a circle. As it sweeps from 0 to 2\pi the state |\psi_\varphi\rangle travels all the way round — and this circle is exactly the equator of the Bloch sphere. The two states we care about sit at opposite ends: |{+}\rangle at \varphi = 0 and |{-}\rangle half a turn away at \varphi = \pi.

Worked example: |+⟩ and |−⟩ fool a measurement

Measure |{+}\rangle in the computational basis: both amplitudes are \tfrac{1}{\sqrt2}, so \Pr(0) = \Pr(1) = \tfrac12. Now measure |{-}\rangle: the amplitudes are \tfrac{1}{\sqrt2} and -\tfrac{1}{\sqrt2}, and squaring kills the minus sign — \Pr(0) = \Pr(1) = \tfrac12 again. Identical statistics. A computational-basis measurement is completely blind to the relative phase: it sees only |\alpha|^2 and |\beta|^2, which are the same for both.

And yet the states are orthogonal — a measurement could tell them apart with certainty, if only it looked the right way. The trick is to change basis first. Apply the Hadamard gate, a gate coming up soon, which maps

H|{+}\rangle = |0\rangle, \qquad H|{-}\rangle = |1\rangle.

Now measure. |{+}\rangle gives 0 every single time; |{-}\rangle gives 1 every single time. The relative phase that was invisible a moment ago has become a perfectly certain, readable bit. Changing basis is how you rotate a hidden phase into an observable outcome — and that conversion is exactly where quantum interference, and the power of quantum algorithms, lives.

Global phase e^{i\gamma}|\psi\rangle is unobservable: it factors out of every probability, so e^{i\gamma}|\psi\rangle and |\psi\rangle are the same physical state (this is the parameter the Bloch sphere drops).
Relative phase \varphi in |0\rangle + e^{i\varphi}|1\rangle is physical: it does not cancel, and different \varphi are genuinely different states (|{+}\rangle and |{-}\rangle are orthogonal).
A computational-basis measurement can't see relative phase (only |\alpha|^2, |\beta|^2); a change of basis (e.g. a Hadamard) turns it into an observable outcome — the seat of interference.

The deep reason is that quantum mechanics predicts only probabilities, and probabilities come from squared magnitudes — an operation that throws away any overall angle. A state is not really a vector; it is a vector up to an overall phase, a "ray" in the mathematician's language. Physicists encode this by describing a state with a density operator |\psi\rangle\langle\psi|, in which the global phase e^{i\gamma} meets its own conjugate e^{-i\gamma} and annihilates — the object literally cannot hold a global phase. Relative phase survives this same test, which is the formal statement of why one is a fiction and the other is fact. If a "global" phase ever did become measurable, you would find it was secretly relative — a phase difference against some other system, like the second qubit in an interference experiment.

This is the single most common phase blunder. The rule "phase doesn't matter, it's unobservable" is true only for a global phase — a factor multiplying the whole state. It is flatly false for a relative phase. Dropping the minus sign in |{-}\rangle = \tfrac{1}{\sqrt2}(|0\rangle - |1\rangle) does not give you "the same state |{+}\rangle" — it hands you an orthogonal, physically distinct state, and any algorithm relying on interference will now give the wrong answer. The tell is where the phase sits: in front of both amplitudes it is global (discard it); in front of one it is relative (keep it, it is real). And beware the reverse trap of thinking a computational-basis measurement "proves" two states are equal because the counts match — |{+}\rangle and |{-}\rangle match perfectly and are still different states.