# How is a qubit represented on a bloch sphere?

A quantum state can be represented as linear combination of 2 states:

In Chuang and Nielsens book, it states that because the squared amplitudes sum to 1:

That the combination can be rewritten as:

My first question is, why exactly is this? And secondly, how is this state represented in the bloch sphere?

I am aware that this may be a simple question to many, but a step by step walkthrough of the algebra underlaying the bloch representation of the qubit would be immensely helpful to me and some friends.

Qubit is described by a vector $$|\psi\rangle = \begin{pmatrix} \alpha \\ \beta \end{pmatrix}$$ where $$\alpha,\beta \in \mathbb{C}$$. So, you would need four real numbers for representing it. Because of condition $$|\alpha|^2 +|\beta|^2 = 1$$, number of degrees of freedom is reduced, so you need only three parameters, hence you can describe the qubit in spacce $$\mathbb{R}^3$$ and you can express it in spherical coordinates. Since two qubits with same global phase are indistinguishable, you need only two parameters for description of the qubit - those are $$\varphi$$ and $$\theta$$.
• @VP9: Assume you have a state $|\psi \rangle$ then state $\mathrm{e}^{i\lambda}|\psi\rangle$ is indistinguishable from $|\psi\rangle$. The parameter $\lambda$ is called global phase. This phase is not represented on Bloch spehere because states $|\psi\rangle$ and $\mathrm{e}^{i\lambda}|\psi\rangle$ are equivalent and therefore they are represented by same vector on Bloch sphere. – Martin Vesely Mar 27 '20 at 5:49