This question is homework from Scott Aaronson's 2017 Quantum Information course.
a) Give an example of a 2×2 unitary matrix where the diagonal entries are 0 but the
off-diagonal entries are nonzero.b) Give an example for a 4×4 unitary matrix.
c) Is it possible to have a 3×3 unitary matrix with this condition? If no, prove it!
(a) is easy.
$$
\begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix}
$$
(c) is not possible. Since for
$$
\begin{pmatrix}
0 & a & b \\
c & 0 & d \\
e & f & 0
\end{pmatrix}
$$
to be unitary the inner product $0\cdot a + c \cdot 0 + e \cdot f$ needs to be zero, but then at least one of $e$ or $f$ needs to be 0.
For (b) I assume he means a 4×4 matrix where the diagonal elements are 0 but all other elements are non-zero (not just any unitary matrix).
I tried without luck:
-
Building the 4×4 unitary matrix from 2×2 unitary block matrices. The closest I got so far is
$$
U = \begin{pmatrix}
X & H \\
H & -X
\end{pmatrix}
$$with
$$
X = \begin{pmatrix}
0 & 1 \\
1 & -0
\end{pmatrix}
$$and
$$
H = \frac{1}{\sqrt 2} \begin{pmatrix}
1 & 1 \\
1 & -1
\end{pmatrix}
$$Unfortunately $U^\dagger U$ leaves some non-diagonal elements.
$$
U^\dagger U = \begin{pmatrix}
2 & 0 & \sqrt 2 & 0 \\
0 & 2 & 0 & \sqrt 2 \\
\sqrt 2 & 0 & 2 & 0 \\
0 & \sqrt 2 & 0 & 2 \\
\end{pmatrix}
$$ -
Starting with an arbitary first column and finding orthogonal column vectors with the desired 0 elements, but this seems tricky.
Any hints?
Best Answer
Example: $$ \frac{1}{\sqrt{3}} \begin{bmatrix} 0 & 1 & 1 & 1 \\ 1 & 0 & 1 & -1 \\ 1 & -1 & 0 & 1 \\ 1 & 1 & -1 & 0 \end{bmatrix}. $$