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In Nielsen and Chuang (page:379), it shows how to represent a 1 qubit depolarizing channel in operator-sum representation. $$ \mathcal{E}_1(\rho)=pI/2+(1-p)\rho =(1-3p/4)\rho+p/4(X\rho X+Y\rho Y+Z\rho Z) $$

How to write a 2 qubit depolarizing channel in operator-sum representation?

$$ \mathcal{E}_2(\rho)=pI/4 +(1-p)\rho $$

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2 Answers 2

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In 2 qubit Hilbert space, $$I_{12}/4$$ can be written in the form $$ I_{12}/4=I_1/2\otimes I_2/2 $$

and $$ I_k/2 =\frac{1}{4} (\rho +X_k\rho X_k+Y_k\rho Y_k+Z_k\rho Z_k) $$

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$$ \mathcal{E}(\rho) = (1 - p) I \otimes I \rho I \otimes I + \sum_{i = 1}^{15} \frac{p}{16} A \otimes B \rho A \otimes B, $$ where $A \otimes B \in \{I, X, Y, Z\} \otimes \{I, X, Y, Z\} \backslash II$

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  • $\begingroup$ This doesn't look trace preserving, you need $(1-15p/16)$ in the first term $\endgroup$
    – forky40
    Commented Sep 2, 2022 at 19:43
  • $\begingroup$ Sure. That's the correct channel though minus the typo. $\endgroup$
    – esabo
    Commented Sep 3, 2022 at 20:09

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