Upper bound on $\Vert U_1 \otimes U_2 \otimes \cdots \otimes U_k - V_1 \otimes V_2 \otimes \cdots \otimes V_k \Vert$

Question

Let $U_i$ and $V_i$ be unitaries that act on the same subsystems. Can we upper bound the difference between the tensor products of these unitaries, i.e. $\Vert U_1 \otimes U_2 \otimes \cdots \otimes U_k - V_1 \otimes V_2 \otimes \cdots \otimes V_k \Vert$, in terms of product of $\Vert U_i - V_i \Vert$? Here, the norm can be any unitarily invariant norm including operator norm, trace norm, and etc.

Answer 1

Let's write $$G_i=U_1\otimes U_2\otimes\ldots U_{i-1}\otimes V_i\otimes V_{i+1}\otimes\ldots\otimes V_k.$$ So, we have $$ \|G_{k+1}-G_1\|=\|G_{k+1}+(G_k-G_k)+\ldots (G_2-G_2)-G_1\|\leq \sum_{i=2}^{k+1}\|G_i-G_{i-1}\|. $$ But, $$ \|G_i-G_{i-1}\|=\|U_{i-1}-V_{i-1}\| $$ So, this is just the same as $$ \|G_{k+1}-G_1\|\leq \sum_{i=1}^{k}\|U_{i}-V_{i}\|. $$