Phase and Global Phase

Overview

Quantum states use complex amplitudes, which means amplitudes can have a phase. Phase is required for quantum computing because it is the mechanism behind interference: phases can cause amplitudes to reinforce or cancel. This page distinguishes relative phase (physically meaningful) from global phase (unobservable).

Intuition

If you have ever added two waves, you have seen phase in action. Two waves that line up add; two waves that are opposite can cancel.

Quantum amplitudes behave like wave-like quantities. The “direction” of a complex number acts like a phase. When you combine contributions to the same outcome, their phases determine whether they add constructively or destructively.

At the same time, not every phase matters. Multiplying the entire state by the same complex factor changes none of the measurement probabilities. This is global phase: it has no direct observable effect.

Formal Description

Write the general qubit state as

$$|\psi\rangle = \alpha|0\rangle + \beta|1\rangle.$$

Any nonzero complex number can be expressed in polar form. Informally, you can think of an amplitude as “magnitude times a phase.”

Global phase. If you multiply the entire state by the same complex number of magnitude 1, such as $e^{i\gamma}$, you get

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

If $|\psi\rangle$ is normalized, then $|\psi'\rangle$ is also normalized. Measurement probabilities in the computational basis do not change because

$$|e^{i\gamma}\alpha|^2 = |\alpha|^2,\quad |e^{i\gamma}\beta|^2 = |\beta|^2.$$

In words: multiplying by a global phase rotates every amplitude by the same amount, but magnitudes stay the same.

Relative phase. The phase difference between the two amplitudes can matter. For example, compare

$$|\psi_+\rangle = \tfrac{1}{\sqrt{2}}(|0\rangle + |1\rangle),\quad |\psi_-\rangle = \tfrac{1}{\sqrt{2}}(|0\rangle - |1\rangle).$$

These differ by a relative sign (a phase of $\pi$ on one component). They produce the same probabilities in the computational basis, but they are different states.

Worked Example

Consider the two states above, $|\psi_+\rangle$ and $|\psi_-\rangle$. If you measure either one immediately in the computational basis, you get 0 and 1 with probability $1/2$.

Now compare that with global phase. Define

$$|\phi\rangle = i\,|\psi_+\rangle.$$

This multiplies both amplitudes by $i$. The probabilities remain exactly the same, and in fact $|\phi\rangle$ is physically the same state as $|\psi_+\rangle$ for the purposes of measurement outcomes.

The key lesson is: global phase never changes probabilities, but relative phase can influence interference when states are combined and compared.

Turtle Tip

Common Pitfalls

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What’s Next

You now have the full conceptual definition of a single-qubit state: basis states, amplitudes, normalization, superposition, and phase. Next we will use this understanding to build a geometric picture of single-qubit states that ignores global phase and focuses on the physically meaningful degrees of freedom.