Phase flip error correction on state $|0\rangle$

Question

When I read documentations about quantum error correction, it generally speak about bit flip error, and phase flip on $|1\rangle$ state, so for example, let's say my initial state is $|\psi\rangle = a|0\rangle+b|1\rangle$, bit flip error correction $|\psi'\rangle = b|0\rangle+a|1\rangle$ or phase flip error correction $|\psi'\rangle = a|0\rangle-b|1\rangle$.

I'm wondering why the phase flip error correction $|\psi'\rangle = -a|0\rangle+b|1\rangle$ is never discussed? And why error correction codes don't correct this phase flip (at least these I have tested)? Is there a reason I'm not aware about ? Maybe this error never happens "in real life"?

Answer 1

The bit- and phase-flip error you are asking for, i.e. $|\psi'\rangle = b|0\rangle-a|1\rangle$ does indeed exist, and error correction for them have been constructed. The phase flip is associated to the $Z$ Pauli matrix and the bit flip to the $X$ pauli matrix. The error you are asking for is associated to the $Y$ pauli matrix. Not that its action is $Y|\psi\rangle=-i(b|0\rangle-a|1\rangle)$, but as it has been stated in the other answers, the global phase has not observational consequences, and such error would be the one you are aksing for.

I recommend this article for a better understanding of the decoherence models. In such you can see discussions about the Pauli channel (a channel with errors $X,Y,Z$) and also an example of a QECC that does indeed correct all of those kinds of errors with weight one.

Answer 2

An error that negates the amplitude of $|0\rangle$ is observationally indistinguishable from an error that negates the $|1\rangle$ state, so there's no need to consider both. By arbitrary historical convention the Z gate negates the $|1\rangle$ state and so things are done in those terms.

Answer 3

The difference between $a|0\rangle-b|1\rangle$ and $-a|0\rangle+b|1\rangle$ is a global phase which has no observational consequences. In other words, to all intents and purposes, those two errors $Z$ and $-Z$ are the same. An error correcting code that corrects for one also corrects for the other (although the global phase may remain).