# What is Qubit? A Qubit is modeled as a vector $(\alpha,\beta)\in\mathbb{C}^2$ s.t. $|\alpha|^2+|\beta|^2=1$. a qubit is represented by $|\psi\rangle:=\alpha\cdot|0\rangle+\beta\cdot|1\rangle$ where $|0\rangle,|1\rangle$ are the standard basis called computational basis. $\alpha,\beta$ are called the amplitude of the qubit. A qubit can be measured and the result is either $|0\rangle$ or $|1\rangle$ with probability of $|\alpha|^2$ for $|1\rangle$ and probability of $|\beta|^2$. >Example: > 1) $\ket\psi=\frac{1}{\sqrt 2} \ket 0+\frac{1}{\sqrt 2} \ket 1$ > 2) $\ket\psi=\frac{\sqrt 2 e^{12i}}{\sqrt 3}\ket 0+\frac{e^{-4i}}{\sqrt 3}\ket 1$ ## Global phase If $\ket\psi=\gamma\ket\phi$ then $|\gamma|=1$ and so we can write it as $\gamma=e^{\theta i}$ for some $\phi\in[0,2\pi]$ . We call $\gamma$ a [[Global phase]]. There is no way to physically distinguish between those 2 qubits and so we can ignore this phase. ## The canonical pure-qubit form Any qubit $|\psi\rangle=\alpha|0\rangle+\beta|1\rangle$ can be formatted as $\boxed{\cos(\frac{\theta}{2})\ket 0+e^{i\phi}\sin(\frac{\theta}{2})\ket1 }$ as follows, let $\alpha=|\alpha|e^{i\gamma},\beta=|\beta|e^{i\delta}$ we have $e^{i\gamma}(|\alpha|+e^{i(\delta-\gamma)}|\beta|)$ $e^{i\gamma}$ is a global phase and can be ignored. compute $\theta$ s.t. $|\alpha|=\cos(\theta/2)$ and $|\beta|=\sin(\theta/2)$ and $\phi=\delta-\gamma$ . A qubit has $\aleph$ possible values of $\alpha,\beta$. It is not possible to "enjoy" this huge representation as measurement collapse to one of two know values. Tags: #tech #math ## Created 2026-02-06 11:34