Bloch Sphere

A Bloch sphere of radius |a| = 1 contains all possible states of a two-state quantum system (qubit). Each Bloch vector fully determines a spin-1/2 density matrix. Used in Exercise Sheet 10 of Statistical Physics by Manfred Salmhofer (2016), available at https://janosh.dev/physics/statistical-physics.

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% A Bloch sphere of radius |a| = 1 contains all possible states of a two-state quantum system (qubit).
% Each Bloch vector fully determines a spin-1/2 density matrix.
% Used in Exercise Sheet 10 of Statistical Physics by Manfred Salmhofer (2016), available at https://janosh.dev/physics/statistical-physics.
\documentclass[tikz]{standalone}
\usetikzlibrary{angles, quotes}
\begin{document}
\begin{tikzpicture}
  % Define radius
  \def\r{3}
  % Bloch vector
  \draw (0, 0) node[circle, fill, inner sep=1] (orig) {} -- (\r/3, \r/2) node[circle, fill, inner sep=0.7, label=above:$\vec{a}$] (a) {};
  \draw[dashed] (orig) -- (\r/3, -\r/5) node (phi) {} -- (a);
  % Sphere
  \draw (orig) circle (\r);
  \draw[dashed] (orig) ellipse (\r{} and \r/3);
  % Axes
  \draw[->] (orig) -- ++(-\r/5, -\r/3) node[below] (x1) {$x_1$};
  \draw[->] (orig) -- ++(\r, 0) node[right] (x2) {$x_2$};
  \draw[->] (orig) -- ++(0, \r) node[above] (x3) {$x_3$};
  % Angles
  \pic [draw=gray, text=gray, ->, "$\phi$"] {angle = x1--orig--phi};
  \pic [draw=gray, text=gray, <-, "$\theta$", angle eccentricity=1.4] {angle = a--orig--x3};
\end{tikzpicture}
\end{document}
 
 
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