Invalid coefficients in Quantum Process Tomography of the Hadamard gate

Question

I am trying to perform QPT for the Hadamard operation as outlined in this document. The basis they use is $$ \{ \rho_{j} \} = \{|0\rangle \langle 0|, |1\rangle \langle 1|, |+\rangle \langle +|, |+i\rangle \langle +i|\}. \tag1$$ In the document (as well as in Nielsen Chuang) they write that we can express any $\mathcal{E}(\rho)$ as a linear combination of the basis states as $$\mathcal{E}(\rho_{j}) = \sum \limits_{k} \lambda_{jk} \rho_{k}\tag2$$ and $$\lambda_{jk}= \mathrm{tr}(\mathcal{E}(\rho_{j})^{\dagger}\rho_{k}).\tag3$$

For $\rho_{0} = |0\rangle \langle 0| $, we get $\mathcal{E}(\rho_{0}) = |+\rangle \langle +|$. Now when I compute the coefficients $\lambda_{0k}$, I get $\lambda_{00} = 0.5, \lambda_{01} = 0.5, \lambda_{02} = 1, \lambda_{03} = 0.5i$ which is clearly is not the right decomposition. I tried it for other matrices as well, but it turned out to be incorrect. What am I doing wrong here?

Answer 1

The formula $(3)$ is only true when the basis $\rho_j$ is orthonormal. Indeed, to derive $(3)$, we hit both sides of $(2)$ with $\mathrm{tr}(.\rho_k)$ and use $\mathrm{tr}(\rho_j^\dagger\rho_k)=\delta_{jk}$ where $\delta_{jk}$ is the Kronecker delta. Anyway, it is straightforward to check that the basis in $(1)$ is not orthonormal.

You can instead use an orthonormal basis, such as \begin{equation} \{|0\rangle\langle 0|,|1\rangle\langle 1|,|0\rangle\langle 1|,|1\rangle\langle 0|\}\tag4 \end{equation} to compute the coefficients $\lambda_{jk}$ and then, if needed, change to the original basis. Alternatively, you can find and use the dual basis of the original one, as discussed for example in this answer.