Suppose we have a one-qubit unitary U that maps |0⟩ to 13|0⟩+2−2i3|1⟩ and |1⟩ to 2+2i3|0⟩−13|1⟩. What is U?
Explicitly indicate multiplication with a * symbol. For instance, you should write 3*i+2 instead of 3i+2.
To determine the unitary operator U, we need to find the matrix representation of U.
Given that U maps |0⟩ to 13|0⟩+2−2i3|1⟩ and |1⟩ to 2+2i3|0⟩−13|1⟩, we can write these mappings as equations:
U|0⟩ = 13|0⟩ + 2−2i3|1⟩
U|1⟩ = 2+2i3|0⟩ − 13|1⟩
Since U is a 2x2 unitary matrix, we can represent it as:
U = |a b|
|c d|
where a, b, c, and d are complex numbers.
Applying the first mapping equation for |0⟩:
U|0⟩ = 13|0⟩ + 2−2i3|1⟩
This can be expanded as:
|a b| |1| = 13|1| + 2−2i3|0|
|0|
This gives two equations:
a = 13
b = 2−2i3
Applying the second mapping equation for |1⟩:
U|1⟩ = 2+2i3|0⟩ − 13|1⟩
Expanding this equation gives:
|a b| |0| = 2+2i3|1| − 13|1|
|1|
This gives two more equations:
c = 2+2i3
d = −13
Putting all the values of a, b, c, and d together, we get:
U = |13 2−2i3|
|2+2i3 −13|
So, the unitary operator U is:
U = |13 2−2i3|
|2+2i3 −13|