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If I have two operators A and B living in the Composite Hilbert Space $H_I \bigotimes H_{II} $ and I want to take the partial trace of $C=AB$ over the subspace $H_I$, i.e., $Tr_I[AB]$, is there any identity that can help me do this in terms of $Tr_I[A]$ and $Tr_I[B]$. Actually what I am interested in is the partial trace of the commutator $[A,B]$.

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Does $AB$ mean $A\otimes B$? – Chris Ferrie Nov 12 '11 at 17:36
up vote 2 down vote accepted

Let $H$ and $K$ be Hilbert spaces with bases $|e_a\rangle$ and $|f_i \rangle$, respectively.

Let $A,B: H \otimes K \to H \otimes K$ be two operators, and let $C=A\circ B$ be their composition. This means that they are of the form

$$ A ~=~|e_a\rangle \otimes |f_i \rangle ~ A^{ai}{}_{bj}~ \langle e^b| \otimes \langle f^j |, $$ $$B ~=~|e_b\rangle \otimes |f_j \rangle ~ B^{bj}{}_{ck}~ \langle e^c| \otimes \langle f^k |, $$ $$ C ~=~|e_a\rangle \otimes |f_i \rangle ~ A^{ai}{}_{bj}~ B^{bj}{}_{ck}~ \langle e^c| \otimes \langle f^k |, $$

where there are implicitly summed over repeated indices. The partial traces over $H$ are

$$ Tr_{H}A~=~ |f_i \rangle ~ A^{ai}{}_{aj}~ \langle f^j|, $$ $$Tr_{H}B ~=~ |f_j \rangle ~ B^{bj}{}_{bk}~ \langle f^k| , $$ $$ Tr_{H}C ~=~ |f_i \rangle ~ A^{ai}{}_{bj}~ B^{bj}{}_{ak}~ \langle f^k|. $$

$Tr_{H}C$ contains in general off-diagonal information, that are not included in $Tr_{H}A$ and $Tr_{H}B$, so $Tr_{H}C$ can in general not be written as a function of $Tr_{H}A$ and $Tr_{H}B$ only.

Similar reasoning applies to the commutator $A\circ B-B\circ A$.

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