The diamond norm distance between two operations is the maximum trace distance between their outputs for any input (including inputs entangled with qubits not being operated on).
Is it the case that the Diamond norm is subadditive under composition?
$$\text{Diamond}(U_1 U_2, V_1 V_2) \stackrel{?}{\leq} \text{Diamond}(U_1, V_1) + \text{Diamond}(U_2, V_2)$$
For arbitrary linear super-operators $U_j$ and $V_j\def\D{\mathrm{Diamond}} \def\Dn#1{\lVert #1 \rVert_\diamond}\def\le{\leqslant} $, we have $$\def\D{\mathrm{Diamond}} \def\Dn#1{\lVert #1 \rVert_\diamond}\def\le{\leqslant} \begin{aligned} \D(U_1 U_2, V_1 V_2) &= \Dn{U_1 U_2 - V_1 V_2} \\&\le \Dn{U_1 U_2 - V_1 U_2} + \Dn{V_1 U_2 - V_1 V_2} \\&= \Dn{(U_1 - V_1) U_2} + \Dn{V_1(U_2-V_2)} \\&\le \Dn{U_1 - V_1} \Dn{U_2} + \Dn{V_1} \Dn{U_2 - V_2}, \end{aligned}$$ writing compositions by juxtaposition for brevity. In the case that $U_j$ and $V_j$ are CPTP maps, we have $\Dn{U_j} = \Dn{V_j} = 1$, so that $$\begin{aligned} \D(U_1 U_2, V_1 V_2) &\le \Dn{U_1 - V_1} \Dn{U_2} + \Dn{V_1} \Dn{U_2 - V_2} \\&= \Dn{U_1 - V_1} + \Dn{U_2 - V_2} \\&= \D(U_1,V_1) + \D(U_2,V_2).\end{aligned}$$