Quantum Computing, exercise 2.2

[Genady]

Which pairs of expressions for quantum states represent the same state? For those pairs that represent different states, describe a measurement for which the probabilities of the two outcomes differ for the two states and give these probabilities.  
(My answers in red.)
a. |0〉 and −|0〉    same  
b. |1〉 and i|1〉    same
c. 1/√2 (|0〉 + |1〉) and 1/√2 (−|0〉 + i|1〉)    different; measure 1/√2 (|0〉 + |1〉); probabilities 1 and 0 
d. 1/√2 (|0〉 + |1〉) and 1/√2 (|0〉 − |1〉)        different; measure 1/√2 (|0〉 + |1〉); probabilities 1 and 0  
e. 1/√2 (|0〉 − |1〉) and 1/√2 (|1〉 − |0〉)      same
f. 1/√2 (|0〉 + i|1〉) and 1/√2 (i|1〉 − |0〉)      different; measure 1/√2 (|0〉 + i|1〉); probabilities 1 and 0
g. 1/√2 (|+〉 + |−〉) and |0〉              same
h. 1/√2 (|i〉 − |−i〉) and |1〉              same
i. 1/√2 (|i〉 + |−i〉) and 1/√2 (|−〉 + |+〉)      same
j. 1/√2 (|0〉 + e^iπ/4 |1〉) and 1/√2 (e^−iπ/4 |0〉 + |1〉) same

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

[Rieffel, Eleanor G.; Polak, Wolfgang H.. Quantum Computing: A Gentle Introduction. The MIT Press.]

 

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