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In Preskill's notes on quantum error correcting codes in Section 7.2, there seems to be no condition on the environment part of the state, i.e. $|0\rangle_E$ in $|\psi\rangle \otimes |0\rangle_E$.

Does it have to belong in a certain Hilbert space of $n$-dimension for the whole discussion to go through, i.e. the Knill-Laflamme condition being necessary and sufficient to perform quantum error correction?

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The Hilbert space of the environment part of the state must have at least as many dimensions as the minimum number of Kraus operators $N$ used to describe the local state's evolution: $$|\psi\rangle\langle\psi|\to\mathrm{Tr}_E[U(|\psi\rangle\langle\psi|\otimes |0\rangle_E\langle 0|)U^\dagger]=\sum_{k=1}^N K_k|\psi\rangle\langle\psi| K_k^\dagger.$$ For the most general possible interaction on a quantum state $|\psi\rangle$ in a Hilbert space of dimension $d$, we need at most $N=d^2$ Kraus operators, so we can always assume $|0\rangle_E$ to be in a Hilbert space of dimension $d^2$ without loss of generality.

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