In the context of Quantum theory of Information, Typical eigenvectors are permutations of basis vectors. Why?

Question

Why is this so? Can someone please give me detailed steps with an explanation?

Answer 1

If I understand your question properly, this is not a general property but it happens to be because we typically choose, in the qubit case for example, operators that anti-commute.

Let $\sigma_1$ and $\sigma_2$ be two such operators that satisfy $\sigma_i^2=I$ and $$ \sigma_1\sigma_2+\sigma_2\sigma_1=0. $$ Since they square to identity, the eigenvalues must be $\pm 1$. Now let $|\psi_{\pm}\rangle$ be the corresponding eigenvectors of $\sigma_1$ with eigenvalues $\pm1$. We see that $$ (\sigma_1\sigma_2+\sigma_2\sigma_1)|\psi_{\pm}\rangle=0=(\sigma_1\mp I)\sigma_2|\psi_{\pm}\rangle. $$ So, $\sigma_2|\psi_{\pm}\rangle$ are the 0 vectors of $\sigma_1\pm I$ respectively, and are hence eigenvectors of $\sigma_1$ with eigenvalues $\mp1$. Thus, $\sigma_2|\psi_{\pm}\rangle=|\psi_{\mp}\rangle$, which is a permutation.

It might help to visualise what's going on on the Bloch sphere. Pick an axis. Let's say the $z$ axis for the sake of argument. We do a rotation such that two rotations returns everything to the same point it started, i.e. we do a rotation half way around the Bloch sphere. So, any point on the equator is mapped to its antipodal point, corresponding to an orthogonal state. Thus, this $z$ rotation acts as a permutation for any orthogonal basis on the equator, such as the $x$ or $y$ axes.

This choice is particularly convenient in quantum information because it means that the bases mutually unbiased, which has useful properties in, for example, cryptography applications.