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# $\mathbb{CP}^1$
**Complex projective line** (or $\mathbb{CP}^1$) is the space in which pure qubit states live. It is defined as the quotient of $\mathbb{C}^2 \setminus \{0\}$ by $\mathbb{C}^\times$.

$$\mathbb{CP}^1 \stackrel{\text{def}}{=} (\mathbb{C}^2 \setminus\{0\})/\mathbb{C}^\times\qquad \mathbb{C}^\times\stackrel{\text{def}}{=}\mathbb{C} \setminus \{0\}$$

## Motivation
To do quantum computing, we usually use the Hilbert space $\mathbb{C}^2$. That's the Hilbert space in which all qubit state vectors (or qubit "kets") $\lvert\psi\rangle$ live in. However, many elements in this space correspond to something unphysical. For example, it contains state vectors whose probabilities don't add up to 100%. There are also elements which we would consider to be duplicates of each other -- up to a global phase.

Here are the 3 major issues with the space $\mathbb{C}^2$

1) The zero vector is in $\mathbb{C}^2$ but it's unphysical. Applying the [[born-rule|Born rule]] yields 0% + 0% = 0%. Meaning, the qubit doesn't even exist. This violates the second axiom of probability which states that probabilities must add up to 100%.
$$\lvert\psi\rangle = \begin{pmatrix}0\\ 0 \end{pmatrix}\qquad \lvert\psi\rangle\in\mathbb{C}^2 \qquad |0|^2 + |0|^2 = 0$$

2) There are many other states which violate the [[born-rule|Born rule]] and are thus considered unphysical. For example, the following is a state which is in state $\lvert 0\rangle$ with 300% probability and in state $\lvert 1\rangle$ with 200% yielding the total probability of 500%. But according to the second axiom of probability, the probabilities must always add up to exactly 100%.
$$\lvert\psi\rangle = \begin{pmatrix}\sqrt{3} \\ \sqrt{2}\end{pmatrix}\qquad \lvert\psi\rangle\in\mathbb{C}^2\qquad |\sqrt{3}|^2 + |\sqrt{2}|^2 = 3 + 2 = 5$$

3) Two states can differ by some phase $e^{i\phi}$ called the [[global-phase|global phase]]. This phase has no effect on measurement outcomes — it is completely unobservable. Therefore, we identify all vectors that differ by a global phase as representing the same physical state.
$$\lvert\psi_1\rangle = \begin{pmatrix}1\\ 0\end{pmatrix}\qquad \lvert\psi_2\rangle = \begin{pmatrix}i \\ 0\end{pmatrix}\qquad \lvert\psi_2\rangle = e^{i\pi/2}\lvert\psi_1\rangle\qquad \lvert\psi\rangle \sim e^{i\phi}\lvert\psi\rangle$$


## Construction
We construct $\mathbb{CP}^1$ by resolving each of these issues individually.

1) We require that the zero vector is not in $\mathbb{CP}^1$
$$\lvert\psi\rangle = \begin{pmatrix}0\\0\end{pmatrix}\qquad \lvert\psi\rangle\notin\mathbb{CP}^1$$

2) If probabilities don't add up to 100% we can always normalize the state vector by scaling it with some real number $A$. We just require $A\neq 0$ since that would turn the state vector into a zero vector which we just removed in 1). Therefore, we identify all vectors that differ by some constant $A\in\mathbb{R}\setminus\{0\}$ i.e. we say they represent the same state. The following is an example for state $\lvert0\rangle$.
$$\lvert 0\rangle \sim \left\{\begin{pmatrix}1\\0\end{pmatrix}, \begin{pmatrix}2\\0\end{pmatrix}, \begin{pmatrix}-1\\ 0\end{pmatrix}, \cdots\right\}\qquad \lvert \psi\rangle\sim A\lvert\psi\rangle\qquad  A\in\mathbb{R}\setminus\{0\}$$

3) We can always factor the global phase $e^{i\phi}$ out of any state vector. Therefore we identify all vectors that differ by some arbitrary global phase $e^{i\phi}$. The following is an example for state $\lvert 1\rangle$.
$$\lvert 1\rangle \sim \left\{\begin{pmatrix}0\\1\end{pmatrix},\begin{pmatrix}0\\i\end{pmatrix}, \begin{pmatrix}0\\e^{i\pi/4}\end{pmatrix}\cdots\right\}\qquad \lvert\psi\rangle\sim e^{i\phi}\lvert\psi\rangle \qquad \phi\in(-\pi,\pi]$$

From 1), we see the overarching space we're looking for is the following. Here, 0 refers to the zero vector in $\mathbb{C}^2$.
$$\mathbb{C}^2 \setminus \{0\}$$

From 2) and 3), we can multiply $A$ and $e^{\phi}$ and get a new complex number $Ae^{i\phi}$ which is an element of complex plane $\mathbb{C}$. However we required $A\neq 0$ in order to avoid the collapsing vectors into a zero vector. Therefore the number $Ae^{i\phi}$ is actually a member of $\mathbb{C}\setminus\{0\}$ otherwise written as $\mathbb{C}^\times$. Here, 0 refers to the zero complex number $0 + i0$. This is sometimes called a "complex plane with a puncture" and it's the space we want to quotient out from 1).

$$\mathbb{C}^\times = \mathbb{C}\setminus\{0\}$$

Putting it all together we constructed the following:

$$\mathbb{CP}^1 = (\mathbb{C^2}\setminus\{0\})/\mathbb{C}^\times\qquad \mathbb{C}^\times = \mathbb{C}\setminus\{0\}$$