Obtaining transition density matrix

I'm trying to assemble a transition density matrix on quantum computer using one-body excitation $\widehat{E}$ operator on 2 active spatial orbitals, i.e. on 4 spin-orbitals. The problem is, I need to rewrite all the four parts of $\widehat{E}$ only via expectation values


I have a one-body excitation operator defined as


$$
\begin{align}
\widehat{E}\_{00} &= \widehat{a}\_0^\dagger \widehat{a}\_0 + \widehat{a}\_2^\dagger \widehat{a}\_2\\
\widehat{E}\_{01} &= \widehat{a}\_0^\dagger \widehat{a}\_1 + \widehat{a}\_2^\dagger \widehat{a}\_3^\dagger\\
\widehat{E}\_{10} &= \widehat{a}\_1^\dagger \widehat{a}\_0 + \widehat{a}\_3^\dagger \widehat{a}\_2\\
\widehat{E}\_{11} &= \widehat{a}\_1^\dagger \widehat{a}\_1 + \widehat{a}\_3^\dagger \widehat{a}\_3
\end{align}
$$


and I need to evaluate a transition density matrix
$$
\begin{bmatrix}
\langle \psi\_0|\widehat{E}\_{00} |\psi\_1 \rangle & \langle \psi\_0|\widehat{E}\_{01} |\psi\_1 \rangle\\
\langle \psi\_0|\widehat{E}\_{10} |\psi\_1 \rangle & \langle \psi\_0|\widehat{E}\_{11} |\psi\_1 \rangle
\end{bmatrix}
$$


without computing $\langle \psi\_i|\widehat{E}\_{ij} |\psi\_j \rangle$ explicitly - as in real-world setting I have no statevectors.


For $\langle \psi\_0|\widehat{E}\_{00} |\psi\_1 \rangle$ and $\langle \psi\_0|\widehat{E}\_{11} |\psi\_1 \rangle$ I'm using the formula from (<https://journals.aps.org/prresearch/pdf/10.1103/PhysRevResearch.1.033062>)


$$\begin{align}
Re\left(\langle \psi\_i | \widehat{U}^\dagger(\theta^\*) \widehat{A} \widehat{U}(\theta^\*)| \psi\_j \rangle \right) &= \langle +^x\_{ij} | \widehat{U}^\dagger(\theta^\*) \widehat{A} \widehat{U}(\theta^\*)| +^x\_{ij} \rangle - \frac{1}{2} \langle \psi\_i | \widehat{U}^\dagger(\theta^\*) \widehat{A} \widehat{U}(\theta^\*)| \psi\_i \rangle - \frac{1}{2} \langle \psi\_j | \widehat{U}^\dagger(\theta^\*) \widehat{A} \widehat{U}(\theta^\*)| \psi\_j \rangle\\
|+^x\_{ij}\rangle &= \frac{|\psi\_i\rangle + | \psi\_j \rangle}{\sqrt{2}},
\end{align}
$$
allowing me to evaluate expectation values on a quantum computer and obtaining the transition amplitude.


**But, while the formula works for $\widehat{E}\_{00}$, $\widehat{E}\_{11}$, it doesn't work for $\widehat{E}\_{01}$, $\widehat{E}\_{10}$. Do you know what's wrong and how could I fix it?**




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### My guess


I'm suspecting, that the problem lies in the fact, that $\widehat{E}\_{00}$ and $\widehat{E}\_{11}$ are Hermitian, while $\widehat{E}\_{01}$ and $\widehat{E}\_{10}$ are not. So, is there any way around this?




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### My guess 2


Well, it seems, that the problematic part (at least from the "proof" what I've seen) is, that it works only with symmetric operators. But, according to the paper it shouldn't be the case.


