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Thanks.
By the way, there is a simple and natural categorical way to think about this:
the trace of an endomorphism is defined in any bradied monoidal category with duals (such as Vect): in string diagram notation you simply draw the morphism as a line and then bend that line around and connect its output to its input.
Now, it that morphism is the endomorphism of a tensor product object , then there is a similarly evident way to "bend around" only the W-strand. That's the partial trace operation.
This is very simple once one draws the diagram, which I won't do right now.
See trace, which should also be interlinked with partial trace
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