- #1
Kreizhn
- 743
- 1
Homework Statement
I need to find a unitary operator that can map two (two-dimensional) pure states [itex] |+\rangle, |-\rangle [/itex] as follows:
[tex] |+\rangle \to \cos\theta |+\rangle + \sin\theta |-\rangle [/tex]
[tex] |-\rangle \to \sin\theta |+\rangle + \cos\theta |- \rangle [/tex]
For an arbitrary angle [itex] 0 \leq \theta \leq \frac\pi4 [/itex]
The Attempt at a Solution
The first obvious attempt at a solution is to simply create a linear system of equations for an element of [itex] U(2) [/itex], and solve, which gives
[tex] \begin{pmatrix} \cos\theta & \sin\theta \\ \sin\theta & \cos\theta \end{pmatrix} [/tex]
However, this is obviously not unitary. Since I know that unitary mappings are not forced to be conformal, I think the mapping does exist, but am unsure where to go from here.