|q⟩
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For
∣
ψ
⟩
=
cos
(
θ
)
∣
0
⟩
+
sin
(
θ
)
∣
1
⟩
|\psi\rangle = \cos(\theta)|0\rangle + \sin(\theta)|1\rangle
∣
ψ
⟩
=
cos
(
θ
)
∣0
⟩
+
sin
(
θ
)
∣1
⟩
with
θ
=
π
/
6
\theta = \pi/6
θ
=
π
/6
, the Z eigenvalues are +1 for
∣
0
⟩
|0\rangle
∣0
⟩
and -1 for
∣
1
⟩
|1\rangle
∣1
⟩
. What is the expectation
⟨
Z
⟩
=
cos
2
(
θ
)
−
sin
2
(
θ
)
\langle Z\rangle = \cos^2(\theta) - \sin^2(\theta)
⟨
Z
⟩
=
cos
2
(
θ
)
−
sin
2
(
θ
)
?
−
1
/
2
-1/2
−
1/2
= -0.5
1
/
2
1/2
1/2
= 0.5
3
/
2
\sqrt{3}/2
3
/2
≈ 0.87
1
/
4
1/4
1/4
= 0.25
Check answer