Note As it has been said in the comments, this definition of Universal-NOT gate seems to differ from others discussed in other posts [1]. This answer uses the definition proposed by the OP, i.e.
ρ=12(I+→b⋅→σ)⟶U(ρ)=12(I−13→b⋅→σ)
Where I use the symbol
I for the identity matrix to avoid confusion with 1 and
→b∈R3 denotes the Bloch vector.
We write the density matrices of the computational basis states explicitly:
ρa=|a⟩⟨a|=12(I+→ba⋅→σ),
where
a∈{0,1}. Expanding this expression readily yields the vectors
ba:
→ba=(0,0,±1).
Applying your definition of
U to these density operators, the action of the operator on basis states can be obtained directly:
U(|0⟩⟨0|)=12(I−13σz)=13|0⟩⟨0|+23|1⟩⟨1|
U(|1⟩⟨1|)=12(I+13σz)=23|0⟩⟨0|+13|1⟩⟨1|
We can observe that, because the factor
1/3 that "damps" the Bloch vector, pure basis states evolve into mixed states; notice that, intuitively, states get closer to the totally mixed state
I/2 if you make the Bloch vector
→b go to zero.
This post imported from StackExchange Physics at 2014-06-11 15:05 (UCT), posted by SE-user Juan Bermejo Vega