Phases as Unit Complex Numbers

In quantum mechanics every amplitude is a complex number, but two amplitudes that differ only by a global phase are physically indistinguishable. What distinguishes them is the relative phase between components of a state. To handle phases cleanly it helps to think of them as points on the unit circle in the complex plane.

The unit circle and Euler's formula

A complex number $z$ lies on the unit circle when its modulus equals one: $|z| = 1$. Every such point can be written

$z = e^{i\theta} = \cos\theta + i\sin\theta,$

where $\theta \in \mathbb{R}$ is the argument (angle measured counter-clockwise from the positive real axis). This identity is Euler's formula, and it is the starting point for almost all phase arithmetic in quantum theory.

A few landmarks on the unit circle are worth memorising:

| $\theta$ | $e^{i\theta}$ | |---|---| | $0$ | $1$ | | $\pi/2$ | $i$ | | $\pi$ | $-1$ | | $3\pi/2$ | $-i$ | | $2\pi$ | $1$ (one full turn) |

Each entry follows directly from $\cos$ and $\sin$ at the corresponding angle.

Phases in quantum states

A single-qubit state is written $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$ with $|\alpha|^2 + |\beta|^2 = 1$. Because $|\alpha|$ and $|\beta|$ are fixed by the probabilities, the remaining freedom lies entirely in the arguments of $\alpha$ and $\beta$. A common parametrisation writes

$|\psi\rangle = \cos\!\left(\tfrac{\theta}{2}\right)|0\rangle + e^{i\phi}\sin\!\left(\tfrac{\theta}{2}\right)|1\rangle,$

where $\phi$ is the relative phase. The factor $e^{i\phi}$ is a unit complex number: it does not change any probability, but it controls how two such states interfere when combined. Interference is the engine behind quantum speedups, so the precise value of $\phi$ matters enormously.

Multiplying phases

Because $|e^{i\theta}| = 1$ for all real $\theta$, the product of any two unit complex numbers is again a unit complex number:

$e^{i\alpha} \cdot e^{i\beta} = e^{i(\alpha + \beta)}.$

Applying two phase shifts in sequence is therefore equivalent to adding their angles. This simple addition rule is what makes the Bloch-sphere picture work. The cleanest way to inject a relative phase $\phi$ is the phase gate $P(\phi) = \mathrm{diag}(1, e^{i\phi})$, which leaves the amplitude of $|0\rangle$ untouched and multiplies the amplitude of $|1\rangle$ by $e^{i\phi}$, shifting the relative phase by exactly $\phi$. The closely related $z$-rotation gate $R_z(\phi) = \mathrm{diag}(e^{-i\phi/2}, e^{i\phi/2})$ spreads the same relative phase $\phi$ symmetrically across the two components; it agrees with $P(\phi)$ up to an overall global phase $e^{-i\phi/2}$, which (being global) is physically irrelevant.

Try it

This is a numerical exercise — your code should return a number. Compute the real part of the unit complex number $e^{i\pi/3}$ (the phase corresponding to an angle of $60°$).

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