  We show that in some cases, catalyst-assisted entanglement transformation
cannot be implemented by multiple-copy transformation for pure states. This
fact, together with the result we obtained in [R. Y. Duan, Y. Feng, X. Li, and
M. S. Ying, Phys. Rev. A 71, 042319 (2005)] that the latter can be completely
implemented by the former, indicates that catalyst-assisted transformation is
strictly more powerful than multiple-copy transformation. For purely
probabilistic setting we find, however, these two kinds of transformations are
geometrically equivalent in the sense that the sets of pure states which can be
converted into a given pure state with maximal probabilities not less than a
given value have the same closure, no matter catalyst-assisted transformation
or multiple-copy transformation is used.
