Amplitude as a Complex Number

The amplitudes in a quantum state vector are complex numbers, not just real ones. Understanding their structure — magnitude and phase — is the key to reading and predicting quantum behaviour.

Complex numbers in one minute

A complex number can be written in rectangular form as

$\alpha = a + b\,i,$

where $a$ is the real part, $b$ is the imaginary part, and $i = \sqrt{-1}$ . Equivalently, the same number has a polar form

$\alpha = r\,e^{i\theta},$

where the magnitude (or modulus) is

$r = |\alpha| = \sqrt{a^2 + b^2}$

and the phase (or argument) is the angle $\theta = \arctan(b/a)$ measured from the positive real axis in the complex plane.

The relationship between the two forms follows from Euler's identity $e^{i\theta} = \cos\theta + i\sin\theta$ , which gives $a = r\cos\theta$ and $b = r\sin\theta$ .

Why quantum amplitudes must be complex

A general single-qubit state is

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

where $\alpha$ and $\beta$ are complex amplitudes. The Born rule says the probability of measuring outcome $0$ is

$P(0) = |\alpha|^2 = \alpha^* \alpha,$

so only the magnitude of an amplitude determines measurement probabilities. The phase of an amplitude — or more precisely the relative phase between two amplitudes — controls interference. When you apply a gate that changes relative phases (for example the $Z$ gate, which sends $|1\rangle \to -|1\rangle$ ), measurement statistics can change dramatically after further gates exploit that phase difference. This is the engine behind most quantum speedups.

Decomposing an example amplitude

Consider $\alpha = \tfrac{1}{2} + \tfrac{\sqrt{3}}{2}\,i$ . Its magnitude is

= \sqrt{\tfrac{1}{4} + \tfrac{3}{4}} = \sqrt{1} = 1.$$ Its phase is $\theta = \arctan\!\left(\tfrac{\sqrt{3}/2}{1/2}\right) = \arctan(\sqrt{3}) = \tfrac{\pi}{3}$, so in polar form $\alpha = e^{i\pi/3}$. An amplitude with unit magnitude like this is called a **phase factor**. Note that a single amplitude having unit magnitude does not mean the state is normalized: normalization requires $|\alpha|^2 + |\beta|^2 = 1$ for the full state vector. What is harmless is multiplying the *entire* state by one global phase $e^{i\varphi}$, since every probability $|\alpha_k|^2$ is unchanged — consistent with the callout above. ## Try it Compute the magnitude of $\alpha = \tfrac{1}{2} + \tfrac{\sqrt{3}}{2}\,i$ and return it as a number. <RunnableExample />