The locking-decoding frontier for generic dynamics: supplemental materials

. (1.2) In the manipulations above we have used the linearity of the superoperator M in the first line. In the second line we have used Lemma 0.4 with γ = I ⊗ σ , noting that |X|= s. The third line follows from the concavity of the square root. We will now use a helpful identity for the trace of an operator squared: TrZ =Tr(Z ⊗ Z)F , where F is defined as follows. Definition 1.1. The swap operator on A⊗2, which is written as A⊗ A, is the unique linear operator F satisfying F (