trouble with a change of basis

Question

I have two O.N sets

$\{|e_i\rangle\}_{i=1}^r$ and $\{|\tilde e_i\rangle\}_{i=1}^r$

Then there is gotta be a change of basis matrix C, such that

$|\tilde e_i\rangle = \sum_{j=1}^rc_{j,i}|e_j\rangle$

I was hopping to use this equation to reduce the following expresion

$|\Psi_{AB}\rangle=\sum_{i=1}^r\tilde s_i^2|\tilde e_i\rangle\langle \tilde e_i|$

into something of the form

$\sum_{i=1}^rs_i|e_i\rangle\langle e_i|$

but I am having trouble with it. all I can do is:

$\sum_{i=1}^r \tilde s_i^2 |e_i\rangle\langle e_i|=\sum_{i=1}^r\sum_{j=1}^r\sum_{j'=1}^r \tilde s_i^2 c_{j,i}c_{j',i}|e_j\rangle\langle e_{j'}|$

If I could reduce $|e_j\rangle\langle e_{j'}|$ to $|e_j\rangle\langle e_{j}|$ the problem would be solved. But I don't think I can do that. Any idea?

Answer 1

It may be impossible to represent $\Psi_{AB}$ without terms of the form $\left|e_j\rangle\langle e_{j'}\right|$ with $j \neq j'$.

For example, take the basis $$\left|\tilde{e}_1\right> = \frac{1}{\sqrt2}(\left|e_1\right> + \left|e_2\right>)$$ $$\left|\tilde{e}_2\right> = \frac{1}{\sqrt2}(\left|e_1\right> - \left|e_2\right>)$$ and take $\Psi = \frac34 \left|\tilde{e}_1\rangle\langle \tilde{e}_1\right| + \frac14 \left|\tilde{e}_2\rangle\langle \tilde{e}_2\right|$. In the other basis we have $$\Psi = \frac12 \left|e_1\rangle\langle e_1\right| + \frac12 \left|e_2\rangle\langle e_2\right| + \frac14 \left|e_1\rangle\langle e_2\right| + \frac14 \left|e_2\rangle\langle e_1\right|.$$