isted one. Furthermore, the mixing of these two has also the same power as pure catalyst-assisted transformation. In other words, any transformation which can be realized collectively on multiple copies and with the aid of a catalyst can be exactly implemented by only providing some appropriate catalyst. Later on, we proved that these two kinds of transformations are asymptotically equivalent in the sense that they can simulate each other’s ability to implement a desired transformation with the same optimal success probability, when the dimension of catalysts and the number of copies provided tend to infinity [@DF05a].

The contribution of the current paper is twofold. First, we show that in some cases catalyst-assisted entanglement transformation is strictly more powerful than multiple-copy one by deriving a sufficient condition when the former cannot be implemented by the latter. Second, for purely probabilistic setting we find, however, these two kinds of transformations are geometrically equivalent. That is, no matter catalyst-assisted transformations or multiple-copy transformations are used, the sets of quantum states that can be converted into a given state with maximal probabilities not less than a given value have the same closure. It is worth noting that the geometrical equivalence between these two kinds of transformations proved in the current paper is different from the asymptotical equivalence shown in [@DF05a]. We will elaborate the difference at the end of Section III after necessary notations have been introduced.

For simplicity, in what follows we denote a bipartite pure state by the probability vector of its Schmidt coefficients. This will not cause any confusion because it is well known that the fundamental properties of a bipartite pure state under LOCC are completely determined by its Schmidt coefficients. Therefore, from now on, we consider only probability vectors (sometimes we even omit the normalization of a nonnegative vector to be a probability one) instead of quantum states and alw