Could someone explain to me, what is the adjoint map of the partial trace map the (tensored with the identity map), or why does the following equality hold? $Tr(C_A\cdot Tr_{B} D_{AB})=Tr((C_A\otimes Id_B)\cdot D_{AB})$? I think I can see it intuitively from wikipedia, but mathematically can someone prove it?
2026-03-04 01:58:30.1772589510
adjoint operator of the partial trace map
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If you write $D=\sum_j X_j\otimes Y_j$, with $X_j\in A$, $Y_j\in B$, then $$ \text{Tr}_B(D)=\sum_j\text{Tr}(Y_j)X_j, $$ and $$ \text{Tr}(C\cdot\text{Tr}(D))=\text{Tr}(\sum_j \text{Tr}(Y_j)CX_j)=\sum_j\text{Tr}(CX_j)\text{Tr}(Y_j). $$ Also, $$ \text{Tr}((C\otimes I)D)=\text{Tr}(\sum_j CX_j\otimes Y_j)=\sum_j\text{Tr}(CX_j)\text{Tr}(Y_j). $$