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For the projector
P
0
=
∣
0
⟩
⟨
0
∣
P_0 = |0\rangle\langle 0|
P
0
=
∣0
⟩
⟨
0∣
and the state
∣
ψ
⟩
=
cos
(
θ
/
2
)
∣
0
⟩
+
sin
(
θ
/
2
)
∣
1
⟩
|\psi\rangle = \cos(\theta/2)|0\rangle + \sin(\theta/2)|1\rangle
∣
ψ
⟩
=
cos
(
θ
/2
)
∣0
⟩
+
sin
(
θ
/2
)
∣1
⟩
with
θ
=
π
/
3
\theta = \pi/3
θ
=
π
/3
, what is the expectation
⟨
ψ
∣
P
0
∣
ψ
⟩
\langle\psi|P_0|\psi\rangle
⟨
ψ
∣
P
0
∣
ψ
⟩
?
1
/
4
1/4
1/4
= 0.25
3
/
2
\sqrt{3}/2
3
/2
≈ 0.87
1
/
2
1/2
1/2
3
/
4
3/4
3/4
= 0.75
Check answer