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Why isn't it obviously

It is nearly yes ? as you need to use linear combinations rather then sums. If $y=\sum_i e_i \otimes b_i$ then $x=yy^* =\sum_{i,j} e_ie_j^* \otimes b_ib_j^*$. This leaves you with two types of summands whose positivity is clear:

(1) $e_ie_i^*\otimes b_ib_i^*$

(2) $e_ie_j^* \otimes b_ib_j^* + e_je_i^* \otimes b_jb_i^*$ whose positivity is clear by the elementary calculation that boils down to multilinearization of $(\alpha e_i+\beta e_j)(\alpha e_i+\beta e_j)^* \otimes (\gamma b_i+\delta b_j)(\gamma b_i+\delta b_j)^*$. (Just write the second summand as a linear combination of these guys.)

Clearly, you need to use subtractions to clear up the summands of type (2).

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