Let us consider a noise channel $\mathcal{E}(\rho)=\sum_i E_i \rho E_i^{\dagger}$, where $\{E_i\}$ are one choice of Kraus operators associated to it.

I assume I will perform error correction and that this channel is correctable. I note the code space $H_C$

Starting from a pure state $|\psi\rangle \in H_C$, we can purify the state after this noise channel as:

$$ |\psi \rangle |b_0 \rangle \to \sum_i E_i |\psi\rangle |b_i\rangle $$

Where $|b_i\rangle$ is an orthonormal basis of some ancillary system used to purify.

I can also consider a basis of orthonormal unitary operator such that $E_i =\sum_{j} \alpha^i_j F_j$. Doing this, I find:

$$\sum_i E_i |\psi\rangle |b_i\rangle= \sum_j F_j |\psi\rangle |\widetilde{b}_j\rangle $$

$$|\widetilde{b}_j\rangle \equiv \sum_i \alpha^i_j |b_i\rangle$$

The new family $|\widetilde{b}_j\rangle$ is now no longer orthonormal in principle.

My question

How is precisely defined an error in the context of quantum error correction. If it appears that $\forall i \neq j \ F_i H_C \perp F_j H_C$ then by measuring in which subspace my system is after the noise channel I can detect and correct it. And in this context it would physically make sense to say that $\{F_i\}$ are representing errors that occured on my system.

Indeed, if my measurement told me "it went in the space $F_k H_C$, I would have the following evolution: from preparation, to noise channel, to measurement outcome:

$$|\psi\rangle |b_0\rangle \to \sum_j F_j |\psi\rangle |\widetilde{b}_j\rangle \to F_k |\psi\rangle |\widetilde{b}_k\rangle $$

It would then make sense to say "the error $F_k$ occured on my system, as one can see from the very last arrow in this equation. This is really what happened on my system (partially because I measured it).

I am not forced to do the hypothesis about $F_j$ being unitaries and leading the system to orthogonal subspaces to be able to do error correction as given from the Knill-Laflamme condition. But, then:

  • Would the operators still being called "errors" ?
  • If so, can we interpret why they are representing errors ? Indeed in my previous explanation it was quite clear that with the hypothesis I made, $F_k$ really represented a well defined error after I measured the system. But in a more general context it is more "abstract". Is there a way to make sense of it ?

A related question on the topic (but different from what I am asking here): Intuition about Knill-Laflamme QEC conditions


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