algebra precalculus - Bounds / Approximation to sum of squares of sum

Can we define any tight upper / lower bound or approximation to the expression,

$\sum_{i = 1}^{N}|x_{i} + y_{i}|^{2}$

in terms of $\sum_{i = 1}^{N} |x_{i}|^{2}$ and $\sum_{i = 1}^{N} |y_{i}|^{2}$, where $x_{i}, y_{i} \in \mathbb{C}, \forall i \in \{1, 2, , \ldots, N\}$.

The bound should be tight enough to represent $\exp \left\{ - \left( \sum_{i = 1}^{N}|x_{i} + y_{i}|^{2} \right) \right\}$ in terms of $\exp \left( - \sum_{i = 1}^{N} |x_{i}|^2 \right)$ and $\exp \left( - \sum_{i = 1}^{N} |y_{i}|^2 \right)$.