Take the 2-minute tour ×
Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. It's 100% free, no registration required.

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 $[A,B]$.

share|improve this question
    
Does $AB$ mean $A\otimes B$? –  Chris Ferrie Nov 12 '11 at 17:36
add comment

1 Answer

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 $A\circ B-B\circ A$.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.