Measurement of a logical operator in a Foliated Surface Code

Question

I am reading Universal Fault Tolerant Measurement Based Quantum Computing. On page 6, under the section Measurement-based qubit transmission, it states that measurement of the logical operators is easily understood by finding logical operators that commute with the measurement operator $M_{1}= \sigma^X_{1}$ by multiplying the logical operators by the stabilizers.

I can see how they get the new logical operators by multiplying them with the stabilizers of the measured qubit. However I am not really sure why this is done, which I think stems from not really understanding the concept of measuring the logical operators either. This also has confused me in surface codes.

Is it done as the new commuting operators will have a common eigenbasis with the measurment operator, and so describe logical operations on the state resulting from the measurement outcome? But if this is the case, why does multiplication by the stabilisers actually give you a set of commuting operations? I can see how this is the case in this instance, but is that the case in general? The stabilisers anti-commute with the measurement in this case. Is it due to both the stabiliser and the logical operation anti-commuting with the measurement, so their product commutes?

Do we always update the set of logical operators after measurement?

Chuang doesn't really expand on them, beyond stating they give logical pauli operations, and Fowler et al defines them, but then doesn't really perform any actions with them that result in their alteration or update.

Edit: Looking at this answer, it seems to state that we update stabilizers, if they have been measured, by multiplying every stabilizer by one of them that anti-commutes with the measurement operator, as these will still be stabilizers of the new state.

Is this why we update the logical operators as well, as the stabilizers $$C[i]=\sigma^{z}[i-1]\sigma^{x}[i]\sigma^{z}[i+1]$$ anti-commutes with $M_{i}= \sigma^X_{i-1}$ ?

Answer 1

Forgot to come back and answer this. By coupling a state into a cluster, we change the logical pauli operators as follows:

$$C_{Z}\sigma^{X}_{1}\to\sigma^{X}_{1}\sigma^{Z}_{2} $$ and $$C_{Z}\sigma^{Z}_{1}\to \sigma^{Z}_{1}$$

From here, we can find an equivalent pair of logical operators by multiplying both of these by the stabilizers $$\sigma^{Z}_{n-1}\sigma^{X}_{n}\sigma^{Z}_{n+1}$$ For example $$\sigma^{Z}_{2}\sigma^{X}_{3}\sigma^{Z}_{4}\sigma^{X}_{1}\sigma^{Z}_{2}\to\sigma^{X}_{1}\sigma^{X}_{3}\sigma^{Z}_{4}$$ and $$\sigma^{Z}_{1}\sigma^{X}_{2}\sigma^{Z}_{3}\sigma^{Z}_{1}\to\sigma^{X}_{2}\sigma^{Z}_{3}$$

Given we are measuring in the $X$ basis, this ensures that the logical operators commute with the measurements on all qubits before the last. The use of ancillas on the last qubit, as outlined in the paper, allows measurement of the whole logical qubit in only the $X$ basis. Now, measuring even qubits in the $X$ basis will give you $Z_{L}$, and odd qubits will give you $X_{L}$.