Let $\mathcal{H}$ be a Hilbert space and let $A_1, A_2, B_1, B_2 \in \mathfrak{B}(\mathcal{H})$ be bounded linear operators. Consider the operators $A_1 \otimes A_2$, $B_1 \otimes B_2$ on $\mathcal{H} \otimes \mathcal{H}$. Let $\| \cdot \|_{\mathfrak{B}(\mathcal{H})}$ denote the operator norm on $\mathfrak{B}(\mathcal{H})$ and $\| \cdot \|_{\mathfrak{B}(\mathcal{H}) \otimes \mathfrak{B}(\mathcal{H})}$ the operator norm on $\mathfrak{B}(\mathcal{H}) \otimes \mathfrak{B}(\mathcal{H})$.
We have directly:
\begin{align} \| A_1 \otimes B_1 &- A_2 \otimes B_2 \|_{\mathfrak{B}(\mathcal{H}) \otimes \mathfrak{B}(\mathcal{H})} \\ &\leq \| A_1 \otimes B_1 - A_1 \otimes B_2 + A_1 \otimes B_2 - A_2 \otimes B_2 \|_{\mathfrak{B}(\mathcal{H}) \otimes \mathfrak{B}(\mathcal{H})} \\ &= \|(A_1 \otimes (B_1 - B_2) + (A_1 - A_2) \otimes B_2 \|_{\mathfrak{B}(\mathcal{H}) \otimes \mathfrak{B}(\mathcal{H})} \\ &\leq \|(A_1 \otimes (B_1 - B_2)\|_{\mathfrak{B}(\mathcal{H}) \otimes \mathfrak{B}(\mathcal{H})} + \|(A_1 - A_2) \otimes B_2 \|_{\mathfrak{B}(\mathcal{H}) \otimes \mathfrak{B}(\mathcal{H})} \\ &= \|A_1\|_{\mathfrak{B}(\mathcal{H})} \| B_1 - B_2\|_{\mathfrak{B}(\mathcal{H})} + \|A_1- A_2\|_{\mathfrak{B}(\mathcal{H})} \| B_2\|_{\mathfrak{B}(\mathcal{H})} \end{align}
Considering the case $A_i = B_i$ with $\|A_i\|_{\mathfrak{B}(\mathcal{H})}= 1$ this becomes (I now omit the indices denoting the norms):
\begin{equation} \| A_1\otimes A_1 - A_2 \otimes A_2\| \leq 2 \| A_1 - A_2\| \end{equation}
Or more general:
\begin{equation} \| \underbrace{A_1\otimes ... \otimes A_1}_\text{n times} - \underbrace{A_2 \otimes ... \otimes A_2}_\text{n times}\| \leq n \|A_1 - A_2\| \end{equation}
Question: Is there any sharper estimate? I.e.
\begin{equation} \| \underbrace{A_1\otimes ... \otimes A_1}_\text{n times} - \underbrace{A_2 \otimes ... \otimes A_2}_\text{n times}\| \leq c_n \|A_1 - A_2\| \end{equation}
with $c_n < n$?