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Knill-Laflamme error correction conditions for correctable error set $\{\hat{E}_j\}$ are $ \hat{P}_{\mathcal{C}} \hat{E}_k^\dagger\hat{E}_l\hat{P}_{\mathcal{C}}\propto\hat{P}_{\mathcal{C}} $ where $\hat{P}_{\mathcal{C}}$ is the projector onto the logical codespace, or $\hat{I}_L$.

And in general, for both correctable and uncorrectable errors, $ \hat{I}_L \hat{E}_k^\dagger\hat{E}_l\hat{I}_L = \epsilon_0\hat{I}_L+\epsilon_X\hat{X}_L+\epsilon_Y\hat{Y}_L+\epsilon_Z\hat{Z}_L $. If the last three terms are non-zero we have uncorrectable errors. For context, I am following Sec IV. Primer: The QEC Matrix of this paper

This was all considering that pure states represent codewords.

Do these conditions change if the codewords are represented by impure(mixed) states with some density operators, say $\hat{\rho}_{0_L}$ and $\hat{\rho}_{1_L}$ such as in real-world experiments?

The codewords should be impure because encoding a qubit into pure states cannot be done with a fidelity $= 1$ in experiments. See for example, this paper that demonstrates beating the break-even point with error correction, using binomial codes.

How is the impurity due to imperfect state preparation handled by error correction conditions? Do we just consider $\hat{I}_L:=\left(\hat{\rho}_{0_L}+\hat{\rho}_{1_L}\right)/\mathcal{N}$ with the normalization $\mathcal{N}$ and check if we can, at least approximately, satisfy $\hat{I}_L \hat{E}_k^\dagger\hat{E}_l\hat{I}_L\propto\hat{I}_L$?

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I answer this question by you.

How is the impurity due to imperfect state preparation handled by error correction conditions?

Assume you have the description of the encoding circuit, $E$, for some code. Given a physical state $|\psi\rangle$, this produces the logical state $|\psi_L\rangle = E(|\psi\rangle)$.

You implement this $E$ on a noisy quantum hardware. Now, it produces noisy state $\rho = E(|\psi\rangle \langle \psi|)$.

You can model this as the circuit $E$ followed by some noise channel $\mathcal{N}$ that distorts $|\psi_L\rangle \langle \psi_L|$ to $\rho$. Therefore, as long as your code is resistant to this kind of noise (can correct the type of errors arising due to this noise), this will be dealt with by the error-correction circuits of your code.

You can continue to use pure code states in your analysis and the Knill-Laflamme conditions continue to hold.

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  • $\begingroup$ Thank you! As a follow-up, what happens if error correction itself is imperfect? Would we then take a similar approach of replacing it with an ideal EC followed by a noise channel? $\endgroup$ Jun 19, 2023 at 8:44
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    $\begingroup$ You will have to learn about fault-tolerant quantum computing to understand this. In short, (a) you use more complicated syndrome measurement circuits to limit errors in the ancilla qubits from spreading to the data qubits, (b) you repeat syndrome measurements multiple times to be sure an error has occurred. $\endgroup$ Jun 19, 2023 at 15:45
  • $\begingroup$ Thank you for the guidance. $\endgroup$ Jun 19, 2023 at 22:46

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