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In Nielsen and Chuang, it is stated that any error is given by a quantum channel with Kraus operators $E_i$. A pure state $\vert\psi\rangle\langle\psi\vert$ becomes $\sum_i E_i\vert\psi\rangle\langle\psi\vert E_i^\dagger$ Assuming we're dealing with qubits, then the Kraus operator can be expressed as a linear combination of Paulis i.e.

$$E_i=e_{i 0} I+e_{i 1} X+e_{i 2} Z+e_{i 3} XZ.$$

Next, it is stated that the unnormalized state $E_i\vert\psi\rangle$ can be decomposed into four terms $|\psi\rangle, X|\psi\rangle, Z|\psi\rangle, XZ|\psi\rangle$ and the error correction is performed individually.

How should one think of this in the density matrix picture? Starting from

$$E_i\vert\psi\rangle\langle\psi\vert E_i^\dagger$$

I would have cross terms, for example, $X\vert\psi\rangle\langle\psi\vert Z$. With these cross terms, it seems like we have two different errors.

It's unclear to me that an error correction code that fixes errors of the form $X\vert\psi\rangle\langle\psi\vert X$ and $Z\vert\psi\rangle\langle\psi\vert Z$ also fixes $X\vert\psi\rangle\langle\psi\vert Z$.

TL;DR how should I convince myself that digitization of errors works when thinking in the density matrix picture?

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2 Answers 2

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I would have cross terms, for example, $X\vert\psi\rangle\langle\psi\vert Z$. With these cross terms, it seems like we have two different errors.

The idea is that your syndrome measurement on some ancilla qubit forces the state to collapse to a specific state $\sigma_i\vert\psi\rangle\langle\psi\vert \sigma_i$ where $\sigma_i$ is a Pauli operator. You can start with an abritrary error but the syndrome measurement forces a collapse into one of these states. Note that you do not have cross terms here.

It's the same as measuring an arbitrary state $\alpha\vert 0 \rangle + \beta\vert 1 \rangle$ in the computational basis - there is no $\vert 0\rangle\langle 1\vert$ outcome after the measurement.

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There's no substitute for just trying it. Do a density matrix simulation of a circuiting encoding the [[5,1,3]] code, performing one of the errors you're worried about, then decoding the code. See if it works.

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