Global versus relative phase

You’ve now seen the Bloch sphere, superposition, and measurement. There’s one subtle piece left — one that confuses almost every beginner and quietly derails their intuition about quantum gates if they don’t get it early.

Look at these two states:

$|\psi_1\rangle = |+\rangle = \frac{1}{\sqrt{2}}\big(|0\rangle + |1\rangle\big)$

$|\psi_2\rangle = e^{i\pi}|+\rangle = \frac{e^{i\pi}}{\sqrt{2}}\big(|0\rangle + |1\rangle\big) = -\frac{1}{\sqrt{2}}\big(|0\rangle + |1\rangle\big)$

They look different. One has a minus sign out front. If you were treating these as classical vectors, you’d say they point in opposite directions. But in quantum mechanics, they are exactly the same state. Not “kind of similar.” Not “indistinguishable by most measurements.” The same.

Now look at these two:

$|\psi_3\rangle = |+\rangle = \frac{1}{\sqrt{2}}\big(|0\rangle + |1\rangle\big)$

$|\psi_4\rangle = |-\rangle = \frac{1}{\sqrt{2}}\big(|0\rangle - |1\rangle\big)$

They also look almost identical — just one minus sign different. But these two are totally different states. Different physical behavior. Different Bloch-sphere position. Different results when fed into the right measurement.

What gives?

Global phase: invisible

A global phase is a complex factor of the form $e^{i\alpha}$ multiplying the entire state vector:

$|\psi'\rangle = e^{i\alpha}|\psi\rangle$

The key fact: global phase is unobservable. No measurement, no gate, no experiment whatsoever can tell $|\psi\rangle$ apart from $e^{i\alpha}|\psi\rangle$. They are the same physical state, full stop.

Why? Because measurements give probabilities $|\text{amplitude}|^2$, and $|e^{i\alpha} \cdot \alpha_0|^2 = |\alpha_0|^2$ — the magnitude of a complex number on the unit circle is 1, so it drops out of the probability. Every outcome is equally likely either way.

This is why we say $|\psi_1\rangle$ and $|\psi_2\rangle$ above are the same. The $-1 = e^{i\pi}$ is a global factor that nothing can see.

Relative phase: very visible

A relative phase is a phase that appears on one component of a superposition but not another:

$|\psi\rangle = \frac{1}{\sqrt{2}}\big(|0\rangle + e^{i\varphi}|1\rangle\big)$

This is not the same as a global phase. Changing $\varphi$ moves the state around the Bloch sphere’s equator. It’s absolutely something you can measure — and it’s the whole reason quantum interference works.

Try it yourself:

Set $\theta = \pi/2$ (equator). Now move $\varphi$ slowly. Watch:

• The Bloch sphere point moves around the equator — different position = different state.
• The amplitude-phase wheels rotate — the clock hand on $|1\rangle$ sweeps around.
• But the probabilities stay stuck at $(0.5, 0.5)$ — in the computational basis, $|+\rangle$ and $|i\rangle$ and $|-\rangle$ all measure 50/50.

The states are genuinely different — but you’d never notice in a measurement along the $z$-axis (which is what the probability bars show). To actually detect the relative phase, you need to do something cleverer: apply a gate that rotates the phase into the measurement direction. A Hadamard will do it. We’ll see this trick in Module 3 when we revisit the gate.

The clock analogy

Think of each amplitude as a little clock hand: its length is the magnitude, its angle is the phase. Here’s the rule:

• Rotating every clock by the same amount (global phase) — nothing observable changes. It’s like agreeing the whole planet moves its clocks forward by 3 hours. The relative timing of events is the same.
• Rotating one clock but not another (relative phase) — everything changes. It’s like your clock moving forward but mine staying put. Now we disagree about when meetings happen.

Every quantum gate is either rotating clocks relative to each other or not. The gates that change relative phase are where the power comes from.

Why this matters for gates

The reason we drag global phases behind the barn and shoot them: they don’t affect anything, and carrying them around in equations is busywork. When we write

$R_x(\pi) = \begin{pmatrix} 0 & -i \\ -i & 0 \end{pmatrix} = -i \cdot X$

that $-i$ out front is a global phase when $R_x(\pi)$ is applied to any state. So for all physical purposes, $R_x(\pi)$ and $X$ are the same gate. We often write "$R_x(\pi) = X$" and quietly drop the global phase.

But if you ever see a circuit that seems to have a “spurious” minus sign somewhere — don’t panic, it’s probably global phase.

What’s next

With this in mind, you’re ready for the rest of single-qubit gates — the Pauli gates, phase gates, and continuous rotations. Global phase will show up occasionally as an inconvenience in the matrix form of a gate; you can always just ignore it.