We investigate entanglement transformations with stochastic local operations
and classical communication (SLOCC) in an asymptotic setting using the concepts
of degeneration and border rank of tensors from algebraic complexity theory.
Results well-known in that field imply that GHZ states can be transformed into
W states at rate 1 for any number of parties. As a generalization, we find that
the asymptotic conversion rate from GHZ states to Dicke states is bounded as
the number of subsystems increase and the number of excitations is fixed. By
generalizing constructions of Coppersmith and Winograd and by using monotones
introduced by Strassen we also compute the conversion rate from W to GHZ
states.Comment: 11 page
