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Both $$\sum_i |i\rangle \langle i | $$ and $$\sum_j |j\rangle \langle j | $$ are summations over basis vectors. The indices $i,j$ run over the same values – values of indices that identify the basis vectors in the same basis (set of vectors) – but the particular values of the indices $i,j$ are independent. Can you calculate how much is the expression below?...
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