This is from page 434 of Nielsen and Chuang:
. Supposing the state of the encoded qubit is |ψ⟩ before the noise acts, then after the noise has acted the state is E(|ψ⟩⟨ψ|). To analyze the effects of error-correction it’s easiest to focus on the effect error-correction has on a single term in this sum, say Ei|ψ⟩⟨ψ|E†i . As an operator on the first qubit alone Ei may be expanded as a linear combination of the identity, the bit flip, the phase flip, and the combined bit and phase flip.The (un-normalized) quantum state Ei|ψ⟩ can thus be written as a superposition of four terms, |ψ⟩, X1|ψ⟩, Z1|ψ⟩, X1Z1|ψ⟩.
Just before, this is stated:
Indeed, the Shor code protects against much more than just bit and phase flip errors on a single qubit – we now show that it protects against completely arbitrary errors, provided they only affect a single qubit! The error can be tiny – a rotation about the z axis of the Bloch sphere by π/263 radians, say – or it can be an apparently disastrous error like removing the qubit entirely and replacing it with garbage!
I understand as to how it's possible for any error with can be defined as an arbitrary gate acting on a single qubit in the system to be corrected, but what does an arbitrary error that yields an un-normalized quantum state even mean? How is it possible for a qubit's resultant state to be un-normalized?