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A quantum state can be represented as linear combination of 2 states: enter image description here

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

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That the combination can be rewritten as:

enter image description here

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

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.

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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 $\theta$.

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  • $\begingroup$ what is global phase? and how do those descriptions become represented on bloch sphere? thank you again Martin . $\endgroup$
    – neutrino
    Mar 27, 2020 at 5:31
  • $\begingroup$ @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. $\endgroup$
    – Martin Vesely
    Mar 27, 2020 at 5:49
  • $\begingroup$ On Bloch sphere, where are magnitude and phase of a qbit? $\endgroup$
    – guest
    Mar 28, 2020 at 3:24

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