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wiki:bloch-sphere [June 13, 2026 at 03:13] – created - external edit 127.0.0.1wiki:bloch-sphere [August 13, 2026 at 18:55] (current) – Ivan Janevski
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 # Bloch sphere # 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. **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.
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 +\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. 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.
  
 $$\lvert\psi\rangle = \cos\frac{\theta}{2}\lvert 0\rangle + e^{i\varphi}\sin\frac{\theta}{2}\lvert 1\rangle$$ $$\lvert\psi\rangle = \cos\frac{\theta}{2}\lvert 0\rangle + e^{i\varphi}\sin\frac{\theta}{2}\lvert 1\rangle$$
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 ## Computational basis states ## Computational basis states
wiki/bloch-sphere.1781320400.md.gz · Last modified: by 127.0.0.1