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I don't have much confidence with density matrices, and I would like to be sure about a property of composition of tensor products operations. Specifically,

$$ \sum_i \sum_j |a_i\rangle|b_i\rangle\langle c_j|\langle d_j| =$$
$$ =\sum_i \sum_j |a_i\rangle \langle c_j| |b_i\rangle \langle d_j| $$
and the reason for this is the following property of the tensor product: $$(x \otimes y)^T(x \otimes y) = (x^Tx \otimes y^Ty) $$

Am I right?

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The simple answer is: yes...$ $

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