Let |ø>=(1−i)/2 |0> −(1+i)/2|1> and |ϕ>=(2+i)/3|0>-2i/3|1>. What is <ø|ϕ>?
(3+5*i)/6
What quantum state do you have to input in order to get output 1/sqrt2(|0>+ |11>)?
If the first qubit is in the state 2/sqrt(5)|0> - 1/sqrt(5)|1> and the second qubit is in the state 1/sqrt(2)|0> - i/sqrt(2)|1>, what is the state of the composite system?
To find the inner product <ø|ϕ>, we need to take the complex conjugate of one of the states and then multiply the corresponding elements together.
Let's start by finding the complex conjugate of |ø> and call it <ø|.
Given |ø> = ((1 - i)/2)|0> - ((1 + i)/2)|1>
To find <ø|, we take the complex conjugate of each coefficient and flip the order of the vector elements:
<ø| = ((1 + i)/2)*<0| - ((1 - i)/2)*<1|
Now, let's find the complex conjugate of |ϕ>:
Given |ϕ> = ((2 + i)/3)|0> - (2i/3)|1>
To find <ϕ|, we take the complex conjugate of each coefficient and flip the order of the vector elements:
<ϕ| = ((2 - i)/3)*<0| + (2i/3)*<1|
Now we have <ø| = ((1 + i)/2)*<0| - ((1 - i)/2)*<1| and <ϕ| = ((2 - i)/3)*<0| + (2i/3)*<1|.
To find <ø|ϕ>, we multiply the corresponding elements together and then sum them up:
<ø|ϕ> = ((1 + i)/2)*((2 - i)/3)*|0><0| + ((1 + i)/2)*(2i/3)*|0><1|
- ((1 - i)/2)*((2 - i)/3)*|1><0| - ((1 - i)/2)*(2i/3)*|1><1|
Simplifying this expression, we get:
<ø|ϕ> = ((1 + i)/2)*((2 - i)/3) - ((1 - i)/2)*(2i/3)
Now we can calculate the value of <ø|ϕ>:
<ø|ϕ> = ((1 + i)/2)*((2 - i)/3) - ((1 - i)/2)*(2i/3)
= (1/2)*(2/3) + (i/2)*(-1/3) + (1/2)*(1/3)i - (i/2)*(2/3) - (i^2/2)*(1/3)i - (1/2)*(2/3)
= 1/3 - i/6 - i/6 + i^2/6
= 1/3 - i/3 + i^2/6
= 1/3 - i/3 - 1/6
= 1/6 - i/3
Therefore, <ø|ϕ> = 1/6 - i/3.