It's easy to take a quantum state represented on the Bloch sphere and rotate it around an arbitrary ray emanating from the origin. On the other hand, we can never use a unitary matrix to get a reflection preserving one coordinate and negating a third. Intuitively, why is this the case? Unitary matrices are isometries on the Bloch sphere, and so are reflections (say reflection over the xy axis), so at first glance it doesn't seem unreasonable that this could be the case.
Asked
Active
Viewed 961 times
1 Answers
5
Intuitively: Because a reflection about the XZ-plane maps Y to -Y, and thus corresponds to a complex conjugation of your state. This is anti-unitary, not unitary, while unitaries are linear maps. (For instance, mapping $x\to\bar x$ is not linear (over the complex numbers), since $\lambda x$ is not mapped to $\lambda \bar x$ for complex $\lambda$.)
Mathematically: Because the isomorphism from SU(2) goes to SO(3), not O(3). Following up on the explanation found there: Reflections are not continuously connected to the identity, while all unitary transformations are continuously connected to the identity (through $U=\exp(iHt)$).
Norbert Schuch
- 22,105