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The observable
A
=
[
1
2
2
1
]
A = \begin{bmatrix}1 & 2 \\ 2 & 1\end{bmatrix}
A
=
[
1
2
2
1
]
has eigenvalues 3 and -1 with normalized eigenvectors
(
1
,
1
)
/
2
(1,1)/\sqrt{2}
(
1
,
1
)
/
2
and
(
1
,
−
1
)
/
2
(1,-1)/\sqrt{2}
(
1
,
−
1
)
/
2
. Using the spectral decomposition
A
=
∑
k
λ
k
∣
v
k
⟩
⟨
v
k
∣
A = \sum_k \lambda_k |v_k\rangle\langle v_k|
A
=
∑
k
λ
k
∣
v
k
⟩
⟨
v
k
∣
, what is the reconstructed (0,0) entry?
1
2 (the average of the eigenvalues)
0
3 (only the
λ
=
3
\lambda=3
λ
=
3
term)
Check answer