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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?

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  • $\begingroup$ The text does not say that the error yields an unnormalized state. They are taking one term from the sum $\sum_i E_i |\psi\rangle \langle \psi | E_i^\dagger$ and analyzing it. Obviously, the entire sum is a normalized density matrix, but each term in it is not. $\endgroup$ Jun 21, 2023 at 6:17
  • $\begingroup$ @AbdullahKhalid This clears it up perfectly. $\endgroup$ Jun 21, 2023 at 8:10

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I think that they did not mean that the state becomes unnormalized. They just meant that the normalization factor cannot be written a superposition of four terms, |ψ⟩, X1|ψ⟩, Z1|ψ⟩, X1Z1|ψ⟩, but since this is only a number it will not affect the error correction procedure.

By the way, if the noise is an erasure channel, the resultant system is in a mixed state, which has an unnormalized density matrix. This still might be corrected if you have well-defined subsystems with well-defined states.

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  • $\begingroup$ I'm not quite sure if I understood the first paragraph correctly. Could you please elaborate on the "not a number" part? $\endgroup$ Jun 21, 2023 at 8:09
  • $\begingroup$ The normalization factor, which is a global factor that does not affect the quantum properties, is not a superposition of four terms, |ψ⟩, X1|ψ⟩, Z1|ψ⟩, X1Z1|ψ⟩. Those four terms are for E1, but you also have E2, E3, etc. The total quantum state is the sum over i on the terms Ei|ψ⟩ = a1|ψ⟩ + b1X1|ψ⟩ + c1Z1|ψ⟩ + d1X1Z1|ψ⟩, and the normalization factor is the square root of SS†, where S is this sum. $\endgroup$ Jun 21, 2023 at 13:32

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