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In the book Quantum computation and quantum information, it says to evaluate $tr(A|\psi\rangle\langle\psi|)$ using Gram-Schmidt procedure to extend $|\psi\rangle$ to an orthonormal basis $|i\rangle$ which includes $|\psi\rangle$ as the first element. Then:

$$tr(A|\psi\rangle\langle\psi|)=\sum_i\langle i|A|\psi\rangle\langle\psi|i\rangle\tag{2.60}$$ $$=\langle\psi|A|\psi\rangle\tag{2.61}$$

I understood that equation 2.61 that uses the special basis $|i\rangle$ described. But in equation 2.60, how $\sum_i\langle i|M|i\rangle$ is correlated to $tr(M)$ ? Can you help me with a more detailed description of it ?

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    $\begingroup$ The equation in bold is just the definition of a trace computed in the computational basis $\{|i\rangle \}$ where $i=1\dots d$ and $d$ is the dimension of the system. You can pick any basis since trace is basis-independent $\endgroup$
    – forky40
    Feb 1, 2021 at 18:30
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    $\begingroup$ Note that by definition: $Tr(M) = \sum_i M_{ii} $ and note that $ \langle i| M | i\rangle = M_{ii} $. $\endgroup$
    – KAJ226
    Feb 1, 2021 at 18:52

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Thanks for the comments, so $tr(M)$ is exactly $\sum_i\langle i|M|i\rangle$ as pointed. I just discovered the same question answered with a proof in the physics stackexchange: https://physics.stackexchange.com/a/104155/273977

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