Let's say I have a pair of states among three parties Alice(A), Bob(B) and Eve(E), $\rho_{ABE}$ and $\rho_{UUE}$ where the first two parties hold uniform values U.}
I know that the trace distance between them is upper bounded by some small quantity $\epsilon$. Like:
$$ ||\rho_{ABE} - \rho_{UUE} ||\le \epsilon . $$ Now, I take the partial trace on B's subsystem, leaving only operators $\rho_{AE}, \rho_{UE}$. What can I say about the trace distance of these two operators? Would the same bound be followed? I.e. is it the case that: $$ ||\rho_{AE} - \rho_{UE} || \le \epsilon. $$