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I was reading the paper https://arxiv.org/abs/1303.4686, where we are given $N$ systems, all with the same Hamiltonian $$H=\sum_i \varepsilon_i \mid i\rangle\langle i\mid ~,$$ such that the joint Hamiltonian is $$h_0 = H\otimes\mathbb{I} \otimes...+\otimes...\otimes +...\mathbb{I}\otimes H ~.$$

The eigenstates of $h_0$ are then given by $\mid \alpha \rangle = \mid i_1 i_2 ... \rangle$ and $\mid \beta\rangle = \mid i_1 i_2 ... \rangle$ ; for example, for N=2, $$\mid \alpha \rangle = \mid 00 \rangle , \mid \beta\rangle = \mid 10\rangle ~.$$

Now, we introduce a unitary operator which is supposed to exchange the populations of these eigenstates--let's call them $P_{\alpha}$ and $P_{\beta}$-- $$U^{\alpha, \beta} = \sum_{n \neq \alpha,\beta}\mid n \rangle\langle n\mid +\mid \alpha \rangle\langle \beta \mid + \mid \beta \rangle\langle \alpha \mid ~,\tag{1} $$ so for our $\mid \alpha \rangle$ and $\mid\beta \rangle$ this would mean $$U^{\alpha, \beta} = \sum_{n \neq \alpha,\beta}\mid n \rangle\langle n\mid +\mid 00 \rangle\langle 10 \mid + \mid 10 \rangle\langle 00 \mid ~.$$

In short, $U^{\alpha, \beta}$ exchanges only the singled-out populations and leaves all others unaffected.

Then the paper makes the claim (equation 6 in the paper) that such a unitary operation (if we control some time dependent potential) can be rewritten as $$U^{\alpha, \beta}(t) = \sum_{n \neq \alpha,\beta} \mid n \rangle\langle n \mid + u^{\alpha\beta}(t)\tag{2}$$ at any moment t during the transposition process, showing that only the affected eigenstates are time dependent, while the rest remain unaltered.

We can then insert eq. (2) into the von Neumann-evolution, $$\rho(t) = U^{\alpha, \beta}(t) \rho U^{\alpha, \beta \dagger}(t)$$ to obtain $$= (P_{\alpha}+ P_{\beta})\rho_1 (t) \otimes \mid 0\rangle\langle 0 \mid + \sum_n P_n \mid n\rangle\langle n\mid ~.\tag{3}$$

The whole point of this calculation was to show that during such 'clever' population exchanges, we can guarantee that, even though the unitary operation is global, it does not create any entanglement.

My question: Is there anyone who can help me derive equation (3)? The authors did not mention how they obtained this result.

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