Why can the most general state of a qubit be written as $|\Psi\rangle=\cos(\frac\theta2)|0\rangle+e^{i\phi}\sin(\frac\theta2)|1\rangle$?

Question

Why we can express a most general qubit as $|\Psi\rangle = \cos{\left(\frac{\theta}{2}\right)}|0\rangle + e^{i \phi} \sin{\left(\frac{\theta}{2}\right)} |1\rangle$? Is there any formal proof for this?

Answer 1

The most general pure state of a qubit can be written as $|\Psi\rangle=a|0\rangle+b|1\rangle$ where $a,b\in\mathbb{C}$. The amplitudes $a$ and $b$ can be written in polar form as $a=re^{i\alpha}$ and $b=se^{i\beta}$ where $r,s\in[0,\infty)$ and $\alpha,\beta\in[0,2\pi)$. Thus, $|\Psi\rangle$ is described by four real parameters $r,s,\alpha,\beta$ as

$$ |\Psi\rangle=re^{i\alpha}|0\rangle+se^{i\beta}|1\rangle.\tag1 $$

However, there are two constraints. First, the squares of the absolute values of the amplitudes are probabilities and therefore sum to one

$$ |a|^2+|b|^2=r^2+s^2=1.\tag2 $$

Consequently, $r,s\in[0,1]$. Now, for any real number $x\in[0,1]$ there exists a unique $\varphi\in[0,\pi)$ such that $x=\cos\frac{\varphi}{2}$. Let $\theta\in[0,\pi)$ be such that $r=\cos\frac{\theta}{2}$. Then from $(2)$ we have that $s=\sin\frac{\theta}{2}$ and substituting into $(1)$, we obtain

$$ |\Psi\rangle=e^{i\alpha}\cos\frac{\theta}{2}|0\rangle+e^{i\beta}\sin\frac{\theta}{2}|1\rangle.\tag3 $$

The second constraint arises from the fact that the global phase is unobservable. This allows us to fix the phase on $|0\rangle$ to be a positive real number. We can force this by dividing the amplitudes in $(3)$ by $e^{i\alpha}$

$$ |\Psi\rangle\equiv\cos\frac{\theta}{2}|0\rangle+e^{i(\beta-\alpha)}\sin\frac{\theta}{2}|1\rangle\tag4 $$

where $\equiv$ denotes equality up to global phase. Defining $\phi=\beta-\alpha$ we have

$$ |\Psi\rangle\equiv\cos\frac{\theta}{2}|0\rangle+e^{i\phi}\sin\frac{\theta}{2}|1\rangle\tag5 $$

where we can take $\phi\in[0,2\pi)$.

Answer 2

We know the general state of a qubit $|\psi\rangle$ is:

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

$$ \alpha, \beta \in \mathbb{C} $$

Since, we cannot measure global phase, we can only measure the difference in phase between the states $|0\rangle$ and $|1\rangle$. Instead of $\alpha$ and $\beta$ to be complex, we can confine them to the real numbers and add a term to tell us the relative phase between them:

$$ |q\rangle = \alpha|0\rangle + e^{i\phi}\beta|1\rangle$$

$$ \alpha, \beta \in \mathbb{R} $$

Now, since the qubits state must be normalised, $$ \sqrt{\alpha^2 + \beta^2} = 1 $$ and using the trigonometric identity:

$$ \sqrt{\sin^2{x} + \cos^2{x}} = 1 $$

to describe the real $\alpha$ and $\beta$ inn terms of one variable $\theta$: $$ \alpha = \cos{\tfrac{\theta}{2}}, \quad \beta=\sin{\tfrac{\theta}{2}} $$

From this we can describe the state of any qubit using the two variables $\phi$ and $\theta$:

$$ |q\rangle = \cos{\tfrac{\theta}{2}}|0\rangle + e^{i\phi}\sin{\tfrac{\theta}{2}}|1\rangle $$

$$ \theta, \phi \in \mathbb{R} $$

If we interpret $θ$ and $ϕ$ as spherical co-ordinates ($r=1$,since the magnitude of the qubit state is
1), we can plot any single qubit state on the surface of a sphere, known as the Bloch sphere.

Example: the state $|+\rangle$ is the same for $\theta=\pi/2$ and $\phi=0$