I was reading up on the quantum depolarizing channel (Preskill's Notes) (stack exchaange explanation), and I'm failing to see how the form

\begin{align} \sigma &= (\mathcal E \otimes \mathbb I)(|\Omega\rangle\langle \Omega|) = \sum_{ij} \mathcal E(|i\rangle\langle j|)\otimes |i\rangle\langle j|\ \end{align}

gets simplified to:

$$ \sigma = \frac{p}{2D}\mathbb I\otimes \sum_{i}|i\rangle\langle i| + (1-p)\frac1D \sum_{ij}|i\rangle\langle j|\otimes |i\rangle\langle j|\ . $$

where then the first term becomes the identity. When I solve the same channel representation, I find the matrix looks like:

$$ \sigma = \frac{p}{2D}\mathbb I\otimes \sum_{ij}|i\rangle\langle j| + (1-p)\frac1D \sum_{ij}|i\rangle\langle j|\otimes |i\rangle\langle j|\ . $$

Where the first term isn't able to simplify into something as simple as the identity, but is of the form $$I_{2} + \sigma^{x}_{2}$$ where the subscript denotes that the operation is on the second qubit.

I can't seem to pinpoint the mistake in my derivation here, so any pointers are much appreciated.



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