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I'm new to Quantum Error Correction, and I have a question on Shor's Code.

If we have a protected subspace, $V \subset \mathbf{C}^2\otimes \cdots \otimes \mathbf{C}^2$

$V=\operatorname{span}\{|0_{l}\rangle, |1_{L}\rangle.$ We also consider Pauli basis of $\mathbf{C}^2\otimes \cdots \otimes \mathbf{C}^2$ of 9 copies, and constructed as follows: Take the basis of $M_2$ consisting of: \begin{eqnarray} \nonumber X=\begin{pmatrix} 0 & 1\\ 1 & 0 \end{pmatrix}, Y= \begin{pmatrix} 0 & i \\ -i & 0 \end{pmatrix}, Z=\begin{pmatrix} 1 & 0\\ 0 & -1 \end{pmatrix} & \text{and} & 1_2. \end{eqnarray} We list the 1-Paulis as $U_1,\cdots ,U_{28}.$ Define the error map as $\mathscr{E}:M_{2^9}\rightarrow M_{2^9}$ by $\mathscr{E}(X)=\frac{1}{28}\sum_{i=1}^{28}U_iXU_i^*$. $\mathscr{E}$ is completely positive and trace preserving. How do we say that it satisfies the Knill Laflamme Theorem and thus ensure the existence of a recovery operator?

I actually got this from Ved Guptas Book on Functional Analysis of Quantum Information Theory it was on Page 60, right after the proposition. Since, the book was primarily math oriented and cause they just introduced this, as an application of Choi's theorem I don't think they talked about Stabilizer Formalism. How would you explicitly show this though? – Anon – 2019-10-21T07:22:07.117

@Anon That's far too broad to address here. Go away and read a bit about the stabilizer formalism, check any relevant questions that you might find on this site, and then ask something more specific. Even if it's more or less this question again, you'd have done some of the correct notational setup and help us to know where to pitch the answer. – DaftWullie – 2019-10-21T07:49:00.303