# Digitization of errors in QEC

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?

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.