  Understanding the nature of multipartite entanglement is a central mission of
quantum information theory. To this end, we investigate the question of
tripartite entanglement convertibility. We find that there exists no easy
criterion to determine whether a general tripartite transformation can be
performed with a nonzero success probability and in fact, the problem is
NP-hard. Our results are based on the connections between multipartite
entanglement and tensor rank (also called Schmidt rank), a key concept in
algebraic complexity theory. Not only does this relationship allow us to
characterize the general difficulty in determining possible entanglement
transformations, but it also enables us to observe the previously overlooked
fact that {\em the Schmidt rank is not an additive entanglement measure}. As a
result, we improve some best known transformation rates between specific
tripartite entangled states. In addition, we find obtaining the most efficient
algorithm for matrix multiplication to be precisely equivalent to determining
the optimal rate of conversion between the Greenberger-Horne-Zeilinger state
and a triangular distribution of three Einstein-Podolsky-Rosen states.
