# What is the actual probability of not losing information (in a depolarizing channel)

The probability that a depolarizing channel doesn't affect the information is usually assumed to be $$1-3p$$, while, for convenience, it is affected with same probability $$p$$ by any Pauli operator $$X,Y,Z$$.

Stepping to the error correction paradigms, it seems to be common to work only with operators $$X$$ and $$Z$$ as they can describe also a $$Y$$ operator (which, however results into a suppressed probability of $$p^2$$).

On contrary if one embeds in the framework also a probability $$p$$ for the $$Y$$ operator, this opens to the possibility, to "enhance" the noiseless probability to $$(1-p)^3 + p^3 = 1-3p + 3p^2.$$

I'm not sure if this is in contrast with the depolarizing channel, which states that the noiseless probability should be only $$1-3p$$. However, I want to argue that such an extra addendum $$3p^2$$ is not in contrast with the channel model, as this states that no noise occurs with probability $$1-3p$$, which doesn't necessarily mean that, if all the noise operators occurs simultaneously, they do not affect the logic of the system, even though a noise occurred.

In conclusion, I wonder whether such a reasoning is correct or I'm missing some fundamental concept regarding channels.