  We consider vector space interference alignment strategies over the $K$-user
interference channel and derive an upper bound on the achievable degrees of
freedom as a function of the channel diversity $L$, where the channel diversity
is modeled by $L$ real-valued parallel channels with coefficients drawn from a
non-degenerate joint distribution. The seminal work of Cadambe and Jafar shows
that when $L$ is unbounded, vector space interference alignment can achieve
$1/2$ degrees of freedom per user independent of the number of users $K$.
However wireless channels have limited diversity in practice, dictated by their
coherence time and bandwidth, and an important question is the number of
degrees of freedom achievable at finite $L$. When $K=3$ and if $L$ is finite,
Bresler et al show that the number of degrees of freedom achievable with vector
space interference alignment is bounded away from $1/2$, and the gap decreases
inversely proportional to $L$. In this paper, we show that when $K\geq4$, the
gap is significantly larger. In particular, the gap to the optimal $1/2$
degrees of freedom per user can decrease at most like $1/\sqrt{L}$, and when
$L$ is smaller than the order of $2^{(K-2)(K-3)}$, it decays at most like
$1/\sqrt[4]{L}$.
