# Bloch sphere
**Bloch sphere** is a geometrical representation of a single qubit state as a point on the surface of a unit sphere in three-dimensional space. It provides an intuitive way to visualize qubit states and the effect of quantum gates as rotations. The Bloch sphere is named after physicist Felix Bloch.



\documentclass{standalone}
\usepackage{tikz}
\usepackage{mathtools}
\usetikzlibrary{arrows.meta}
\begin{document}
\begin{tikzpicture}[line cap=round, line join=round, >=Triangle]
  \clip(-2.19,-2.49) rectangle (2.66,2.58);
  \draw [shift={(0,0)}, lightgray, fill, fill opacity=0.1] (0,0) -- (56.7:0.4) arc (56.7:90.:0.4) -- cycle;
  \draw [shift={(0,0)}, lightgray, fill, fill opacity=0.1] (0,0) -- (-135.7:0.4) arc (-135.7:-33.2:0.4) -- cycle;
  \draw(0,0) circle (2cm);
  \draw [rotate around={0.:(0.,0.)},dash pattern=on 3pt off 3pt] (0,0) ellipse (2cm and 0.9cm);
  \draw (0,0)-- (0.70,1.07);
  \draw [->] (0,0) -- (0,2);
  \draw [->] (0,0) -- (-0.81,-0.79);
  \draw [->] (0,0) -- (2,0);
  \draw [dotted] (0.7,1)-- (0.7,-0.46);
  \draw [dotted] (0,0)-- (0.7,-0.46);
  \draw (-0.08,-0.3) node[anchor=north west] {$\varphi$};
  \draw (0.01,0.9) node[anchor=north west] {$\theta$};
  \draw (-1.01,-0.72) node[anchor=north west] {$\mathbf {\hat{x}}$};
  \draw (2.07,0.3) node[anchor=north west] {$\mathbf {\hat{y}}$};
  \draw (-0.5,2.6) node[anchor=north west] {$\mathbf {\hat{z}=|0\rangle}$};
  \draw (-0.4,-2) node[anchor=north west] {$-\mathbf {\hat{z}=|1\rangle}$};
  \draw (0.4,1.65) node[anchor=north west] {$|\psi\rangle$};
  \scriptsize
  \draw [fill] (0,0) circle (1.5pt);
  \draw [fill] (0.7,1.1) circle (0.5pt);
\end{tikzpicture}
\end{document}


A single qubit $\lvert\psi\rangle = a\lvert 0\rangle + b\lvert 1\rangle$ has two complex probability amplitudes $a, b \in \mathbb{C}$, subject to the normalization constraint $|a|^2 + |b|^2 = 1$. Because global phase is physically unobservable, the qubit is fully described by just two real parameters: a polar angle $\theta \in [0, \pi]$ and an azimuthal angle $\varphi \in [0, 2\pi)$. We can therefore write any pure qubit state in the following standard form.

$$\lvert\psi\rangle = \cos\frac{\theta}{2}\lvert 0\rangle + e^{i\varphi}\sin\frac{\theta}{2}\lvert 1\rangle$$



## Computational basis states
The two computational basis states $\lvert 0\rangle$ and $\lvert 1\rangle$ sit at the north and south poles of the Bloch sphere respectively. The state $\lvert 0\rangle$ corresponds to $\theta = 0$ (north pole, coordinates $(0,0,1)$), while $\lvert 1\rangle$ corresponds to $\theta = \pi$ (south pole, coordinates $(0,0,-1)$). The six cardinal states — $\lvert 0\rangle$, $\lvert 1\rangle$, $\lvert +\rangle$, $\lvert -\rangle$, $\lvert i\rangle$, $\lvert -i\rangle$ — sit at the six poles of the three coordinate axes.

## Bloch vector
Any qubit state $\lvert\psi\rangle$ maps to a point on the surface of the unit sphere. The Cartesian coordinates $(x, y, z)$ of the corresponding Bloch vector are obtained from the probability amplitudes via the [[hopf-fibration]].

$$x = 2\,\mathfrak{Re}(a^* b), \qquad y = 2\,\mathfrak{Im}(a^* b), \qquad z = |a|^2 - |b|^2$$

These coordinates satisfy $x^2 + y^2 + z^2 = 1$, confirming that every pure state lies on the surface of the unit sphere. Mixed states occupy the interior of the sphere, where $x^2 + y^2 + z^2 < 1$.

## Quantum gates as rotations
Single-qubit gates act on the Bloch sphere as rotations. The rotation gates $R_x(\theta)$, $R_y(\theta)$, $R_z(\theta)$ rotate the Bloch vector by angle $\theta$ around the $x$-, $y$-, and $z$-axes respectively. The Pauli gates $X$, $Y$, $Z$ are $\pi$ rotations about the corresponding axes. The Hadamard gate $H$ corresponds to a $\pi$ rotation about the axis halfway between $x$ and $z$, mapping $\lvert 0\rangle \leftrightarrow \lvert +\rangle$ and $\lvert 1\rangle \leftrightarrow \lvert -\rangle$.