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Is there an expression for the partial trace of vectorized density matrix? I did some literature review but didn't find not much relevant information.

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Let $\rho$ be a bipartite linear operator (it doesn't really matter here whether it's Hermitian, positive, or anything else). Denote with $\operatorname{vec}(\rho)$ its vectorisation.

The partial trace of $\rho$ equals, in components: $$\operatorname{Tr}_2(\rho) = \sum_{ij} |i\rangle\!\langle j| \, \sum_k \rho_{ik,jk}.$$ In terms of the vectorisation, you could write this as $(I\otimes \langle m|)\operatorname{vec}(\rho)$ with $|m\rangle\equiv \sum_k |k,k\rangle$. You can verify directly that this gives the correct expression for $\operatorname{vec}(\operatorname{Tr}_2(\rho))$.

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  • $\begingroup$ Did your vec mean stacks up rows? With dirac notation, $vec\left( |i\rangle \langle j| \right) =|i\rangle |j\rangle ,vec\left( |ij\rangle \langle kl| \right) =|ij\rangle |kl\rangle $? $\endgroup$
    – narip
    Oct 1, 2022 at 11:43
  • $\begingroup$ @narip yes that's what I mean. I find that notation convenient because it maps matrices to vectors with the same components (in the standard conventions): $\rho_{ij}=(\operatorname{vec}(\rho))_{ij}$ $\endgroup$
    – glS
    Oct 1, 2022 at 22:35
  • $\begingroup$ But shouldn't $I\otimes \langle m|vec\left( \rho \right) =I\otimes \sum_k{\langle kk|}\sum_{ijmn}{\rho _{ij,mn}|ij\rangle |mn\rangle}=\sum_{ijk}{\rho _{ij,kk}|ij\rangle}$? $\endgroup$
    – narip
    Oct 2, 2022 at 0:15
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    $\begingroup$ @narip uhm, good point. The operators are $\rho:A\otimes B\to C\otimes D$, while $\operatorname{vec}(\rho)\in (C\otimes D)\otimes(A\otimes B)$. When writing $(I\otimes\langle m|){\rm vec}(\rho)$ we think of ${\rm vec}(\rho)$ as a vector in the bipartite space with above structure. Thus here $I\equiv I_{CD}$, and $\langle m|$ acts on $A\otimes B$, which gives the incorrect result, as you noted. I instead need $\langle m|$ to act on the space $D\otimes B$. Formally, I think we can fix this having a swap act between $D$ and $A$ before the projection $\endgroup$
    – glS
    Oct 2, 2022 at 0:28
  • $\begingroup$ @narip I think the composite density matrix should be $|ik\rangle \langle jl|$ if you use $|i \rangle\langle j|$ for the single system $\endgroup$
    – Will Yang
    Oct 10, 2022 at 19:08

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