# How does one compute $(\text{id}\otimes T)\rho$ for a possibly entangled quantum state $\rho$?

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?

• 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
– glS
Commented Mar 5, 2018 at 18:45
• 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 $(\text{id}\otimes T)(|0,0\rangle\langle 0,0|)=|0,\psi\rangle\langle 0,\psi|$? Commented Mar 5, 2018 at 20:43
• 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
– glS
Commented Mar 5, 2018 at 20:53

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.