# Knill Laflamme conditon

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?

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.