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Qubits
12. Calculate the 16 elements of the following direct product in symbolic form:
(HINT. First element would be ae, second element would be af
...
)
2
4
3
5
2
4
3
5 ¼ ?
a
b
c
d
e
f
g
h
13. Calculate the four elements of the following matrix combination in symbolic
form: (HINT. Element (1,1) would be ae + bg + w)
ef
gh
ab
cd
wx
yz
þ
¼ ?
14. Prove that the 2 2 Hadamard matrix H is its own inverse H 1 . (HINT: Show
that HH 1
¼ 1 , a diagonal matrix with ones on the diagonal)
11
1
H
¼
1
15. Provide a normalized state vector of dimension four with 70 % chance for the
first given combination to be observed, and 10 % chance for the remaining
three combinations to be observed. (ANS. [0.8366 0.316 0.316 0.316] 0 )
16. Assume
an entangled state vector with two qubits of
the
form
> Þ 0
ψ >¼ ηð
0
>
0
> þ
1
>
1
(a) When this vector is observed, what is the probability of a zero in the first
qubit? (ANS. 50 %)
(b) If a zero is observed in the first qubit, what is the probability of a zero in the
second qubit? (ANS. 100 %)
(c) Assume the qubits are physically separated and that the second qubit is
taken to the other side of the world. Assume an observation of the first qubit
indicated that it was a one. What is the probability of seeing a one for the
second qubit when it is observed? (ANS. 100 %)
(d) What are the conditions on these physically separated qubits? (ANS. They
must continue to be in a coherent quantum mechanical system and thus
continue to be entangled)
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