Quantum Computing, exercise 2.3

[Genady]

From Rieffel, Eleanor G.; Polak, Wolfgang H.. Quantum Computing: A Gentle Introduction:

Which states are superpositions with respect to the standard basis, and which are  not? For each state that is a superposition, give a basis with respect to which it is not a superposition.  
a. |+〉  
b. 1/√2 (|+〉 + |−〉)  
c. 1/√2 (|+〉 − |−〉)  
d. √3/2 |+〉 − 1/2 |−〉  
e. 1/√2 (|i〉 − |−i〉)  
f. 1/√2 (|0〉 − |1〉)

 

*Definitions:
|+〉 = 1/√2 (|0〉 + |1〉)
|−〉 = 1/√2 (|0〉 − |1〉)
|i〉 = 1/√2 (|0〉 + i|1〉)
|−i〉 = 1/√2 (|0〉 −i|1〉)

 

My answers:

a. superposition; basis: |+〉, |−〉
b. not a superposition
c. not a superposition
d. superposition; basis: √3/2 |+〉 − 1/2 |−〉, 1/2 |+〉 + √3/2 |−〉
e. not a superposition
f. superposition; basis: |+〉, |−〉

 

Agree?

[Genady]

Exercise 2.4:

Which of the states in 2.3 are superpositions with respect to the Hadamard basis (|+〉, |−〉), and which are not?

 

My answer:

Superpositions: b, c, d, e.

Not superpositions: a, f.

 

OK?

[Genady]

Next:

 

My answers:

a. π

b, c. any

[Genady]

Continuing the same theme:

 

My answers:

a. 3/4, 1/4

b. 1/4, 3/4

c. 1/2, 1/2

d. 1/2, 1/2

e. 1/2, 1/2

f. 1/2, 1/2

g. 1/8*(4+2√3), 1/8*(4-2√3)

Does it look right?

I am not sure what they mean "describe possible measurement outcomes". I think that the outcomes are just Yes or No for whatever is measured each time.

[Genady]

The last one on this theme:

 

My answers:

a. 1/√2 (|0> - i|1>)

b. 1/√2 (|0> - i|1>)

c. 1/√2 (|0> - e^iπ/6|1>)

d. √3/2 |+> + i/2 |- >

 

I am not sure what they mean by "all". I think that for each state there is only one orthonormal basis that includes that state, up to a phase.

(In this case of the two-dimensional states, of course.)