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I consider measuring Pauli operator m in either normal stabilizer codes and subsystem codes. In normal stabilizer codes, if m commutes with the stabilizer group, measurement outcome is deterministic (1 or -1 depending on whether m or -m is in the stabilizer group). On the other hand, if m anti-commutes with some stabilizer operators, the measurement outcome is 50/50 random. This randomness is derived by calculating the probabilities of projective measurement using the fact that m anti-commutes with some stabilizer operators.

Now, I am confused about the probability of obtaining measurement outcome when performing gauge fixing in subsystem codes. I understand that gauge operators is not in the stabilizer group, and the state is not eigenstate of gauge operators, so the eigenvalues of gauge operators are not fixed until we measure the gauge operators. In this sense, the measurement outcome of a gauge operator seems 50/50 random. However, since a gauge operator commutes with the stabilizer group, we can not use the derivation that we used in the case of normal stabilizer codes above, so I cannot understand whether it is actually random or not.

In summary, the fact that an eigenvalue of the gauge operator is not fixed until we measure it, so the outcome seems 50/50 random and the fact that a gauge operator commutes with the stabilizer group, so the outcome does not seem 50/50 random are contradictory to me. What am I misunderstanding?

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