How are eigenvectors and eigenvalues expressed in the Bloch sphere? - Quantum Computing Stack Exchange most recent 30 from quantumcomputing.stackexchange.com 2022-01-20T16:38:55Z https://quantumcomputing.stackexchange.com/feeds/question/13282 https://creativecommons.org/licenses/by-sa/4.0/rdf https://quantumcomputing.stackexchange.com/q/13282 1 How are eigenvectors and eigenvalues expressed in the Bloch sphere? ts549 https://quantumcomputing.stackexchange.com/users/12915 2020-08-12T13:30:18Z 2020-11-09T17:49:11Z <p>I'm relatively new to the subject of quantum computing, and I recently came across the idea of eigenvalues and eigenvectors. I believe I understand the relationship between the two, where eigenvalues determine the factor by which an eigenvector is stretched, and eigenvectors represent the direction that the transformation is pointing towards, but I was wondering how exactly eigenvectors and eigenvalues would be expressed in the Bloch sphere.</p> https://quantumcomputing.stackexchange.com/questions/13282/-/13283#13283 1 Answer by DaftWullie for How are eigenvectors and eigenvalues expressed in the Bloch sphere? DaftWullie https://quantumcomputing.stackexchange.com/users/1837 2020-08-12T14:04:53Z 2020-08-12T14:04:53Z <p>The eigenvectors for a one-qubit unitary are two orthogonal vectors. As such, on the Bloch sphere, they are visualised as a single axis (going through the origin). (Remember that angles on the Bloch sphere are doubled so orthogonal states are an angle <span class="math-container">$\pi$</span> different on the Bloch sphere, i.e. opposite directions along the same axis.)</p> <p>The eigenvalue (or, more precisely, the relative angle between the two eigenvalues) is the angle of rotation around that axis.</p> https://quantumcomputing.stackexchange.com/questions/13282/-/13285#13285 0 Answer by glS for How are eigenvectors and eigenvalues expressed in the Bloch sphere? glS https://quantumcomputing.stackexchange.com/users/55 2020-08-12T14:56:34Z 2020-08-12T15:16:04Z <p>A state <span class="math-container">$\rho$</span> with Bloch sphere coordinates <span class="math-container">$\newcommand{\bs}{\boldsymbol{#1}}\bs r\equiv (x,y,z)$</span> has the form <span class="math-container">$$\rho = \frac{I + \bs r\cdot\bs \sigma}{2}\equiv \frac{I+x\sigma_x + y \sigma_y + z\sigma_z}{2},$$</span> with <span class="math-container">$\sigma_x,\sigma_y,\sigma_z$</span> the Pauli matrices.</p> <p>Computing the eigenvalues (eigenvectors) of <span class="math-container">$\rho$</span> thus amounts to computing those of <span class="math-container">$\bs r\cdot\bs\sigma$</span>. Observe that <span class="math-container">$$\bs r\cdot\bs \sigma=\begin{pmatrix}z &amp; x-iy \\ x+iy &amp; -z,\end{pmatrix}$$</span> and thus the eigenvalues are <span class="math-container">$\lambda_\pm = \pm\sqrt{-\det(\bs r\cdot\bs \sigma)}=\pm\|\bs r\|$</span>. The corresponding eigenvectors are then seen to be <span class="math-container">$$\lvert\lambda_\pm\rangle = \frac{1}{\sqrt{2\|\bs r\|(\|\bs r\|\mp z)}}\begin{pmatrix}x-iy \\ \pm \|\bs r\| - z\end{pmatrix}.$$</span> The vectors in the Bloch sphere corresponding to <span class="math-container">$\lvert\lambda_\pm\rangle$</span> have coordinates <span class="math-container">$$\begin{cases} x_\pm &amp;=&amp; \pm x/ \|\bs r\|, \\ y_\pm &amp;=&amp; \pm y/ \|\bs r\|, \\ z_\pm &amp;=&amp; \pm z/ \|\bs r\|. \end{cases}$$</span> In other words, the eigenvectors of <span class="math-container">$\bs r\cdot\bs\sigma$</span> correspond to the two unit vectors in the Bloch sphere along the same direction as <span class="math-container">$\rho$</span>.</p> <p>The eigenvectors of <span class="math-container">$\rho$</span> are then clearly the same as those of <span class="math-container">$\bs r\cdot\bs \sigma$</span>, while its eigenvalues are <span class="math-container">$(1\pm\lambda_\pm)/2$</span>.</p>