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Suppose that $T$ is a quantum channel on a marginal system. How does one compute $(\text{id}\otimes T)\rho$ for a possibly entangled quantum state $\rho$ on the composite system?

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  • $\begingroup$ It's not clear what you are asking. Can you provide some context, and even better an example of the kind of $\rho$ and $T$ you are looking at? Otherwise I would just say that you compute that product the same way you compute any product between two matrices $\endgroup$
    – glS
    Commented Mar 5, 2018 at 18:45
  • $\begingroup$ Well, take for instance $T$ defined on the system $B$ by $T(\rho)=\text{tr}(\rho)\sigma$ for a fixed density operator sigma. Is it true if $\sigma=|\psi\rangle\langle\psi|$, then $(\text{id}\otimes T)(|0,1\rangle\langle 1,0|)=0$ whilst $(\text{id}\otimes T)(|0,0\rangle\langle 0,0|)=|0,\psi\rangle\langle 0,\psi|$? $\endgroup$
    – julian
    Commented Mar 5, 2018 at 20:43
  • $\begingroup$ you can add such examples to your question by editing the post. Also, in what spaces do $\rho$ and $\sigma$ live in your example? And is tr a partial or a total trace? Finally, why can't you just use the regular matrix multiplication rules? Explain what exactly you don't find clear $\endgroup$
    – glS
    Commented Mar 5, 2018 at 20:53

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Late to the party obviously, but for any matrix (not necessarily state) $\rho$ on the composite system there exist matrices $\rho_{jk}$ on the second system such that $\rho=\sum_{j,k}|j\rangle\langle k|\otimes\rho_{jk}$. Indeed, using the usual Kronecker product $$ A\otimes B=\begin{pmatrix}a_{11}B&\cdots&a_{1n}B\\\vdots&\ddots&\vdots\\a_{n1}B&\cdots&a_{nn}B\end{pmatrix} $$ these $\rho_{jk}$ are just the corresponding blocks (here: $B$) in the $j,k$-th position of the big matrix $\rho$. With this in mind---using linearity---any linear map $T$ on the second system acts locally via \begin{align*} ({\rm id}\otimes T)(\rho)&=({\rm id}\otimes T)\Big( \sum_{j,k}|j\rangle\langle k|\otimes\rho_{jk} \Big)\\ &=\sum_{j,k}{\rm id}(|j\rangle\langle k|)\otimes T(\rho_{jk})\\ &=\sum_{j,k}|j\rangle\langle k|\otimes T(\rho_{jk})\\ &=\begin{pmatrix}T(\rho_{11})&\cdots&T(\rho_{1n})\\\vdots&\ddots&\vdots\\T(\rho_{n1})&\cdots&T(\rho_{nn})\end{pmatrix}\,. \end{align*} In other words $T$ is applied to each sub-block of $\rho$ individually.

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