  Given a pair of elliptic curves $E_1$ and $E_2$ over the rational field
$\mathbb Q$ whose $j$-invariants are not simultaneously 0 or 1728, Kuwata and
Wang proved the existence of infinitely many square-free rationals $d$ such
that the $d$-quadratic twists of $E_1$ and $E_2$ are both of positive rank. We
construct infinite families of pairs of elliptic curves $E_1$ and $E_2$ over
$\mathbb Q$ such that for each pair there exist infinitely many square-free
rationals $d$ for which the $d$-quadratic twists of $E_1$ and $E_2$ are both of
rank at least 2.
