don’t spread to uncorrectable errors during the error correction protocol. Note however that the first condition alone isn’t sufficient. For instance, the trivial protocol where no correction is ever applied at the end of the EC round also satisfies the first condition, but clearly is not fault-tolerant.

The second condition is not always checked for protocols in the literature, but it is important as it ensures that errors do not accumulate uncontrollably in consecutive rounds of error correction (see [@AGP06] for a rigorous proof and [@CDT09] for an analysis of the role of input errors in an extended rectangle). To give further motivation as to why the second condition is important, consider a scenario with $s$ faults introduced during each round of error correction, and assume that $t/n<s<(2t+1)/3$ for some integer $n$ (see Fig. \[fig:ConditionTwoJustification\]). Consider an error correction protocol in which $r$ input errors and $s$ faults in an EC block leads to an output state with at most $r+s$ errors[^1]. Clearly condition 1 is satisfied.

With the above considerations, an input state $E_{1}{|\bar{\psi}\rangle}$ with $\text{wt}(E_{1})\leq s$ is taken to $E_{2}{|\bar{\psi}\rangle}$, with $\text{wt}(E_{2})\leq 2s$ by one error correction round with $s$ faults. After the $j$th round, the state will 