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I have been working on a question where I have to decompose this matrix in terms of Pauli Matrices: \begin{bmatrix}1&0&0&1\\0&0&0&0\\0&0&0&0\\1&0&0&1\end{bmatrix}

I already have a solution but I don't understand the solution I've been given, this is the solution:

First there is a truth table:

```
+------+-----------+
|Input | Output |
+------+-----------+
| |00> | |00>+|11> | = 1/2(|00>+|11>+|11>+|11>)
| |01> | 0 | = 1/2(|01>-|01>+|10>-|10>)
| |10> | 0 | = 1/2(|10>-|10>+|01>-|01>)
| |11> | |11>+|00> | = 1/2(|00>+|00>+|11>+|11>)
+------+-----------+
```

I understand the truth table, but I don't understand the things after the "=" and I also don't understand how the final answer is achieved. This is the final answer:

$$ \frac{1}2(I_1 \otimes I_2) + \frac{1}2(Z_1 \otimes Z_2) + \frac{1}2(X_1 \otimes X_2) - \frac{1}2(Y_1 \otimes Y_2) $$

Any help in understanding the solution would be really appreciated. Thank you!

Does this answer your question? Example of Hamiltonian decomposition into Pauli matrices

– user1271772 – 2021-02-27T21:20:13.850