Local Transformations of Multiple Multipartite States

We are grateful to the referee for their time and useful feedback. Taking their remarks into account (which we copy here for completeness), we made the following modifications to our submission.

concerning the motivation: The authors motivate their study with potential applications for condensed matter systems as well as with the usage of quantum memories. In both cases, however, mixed states will inevitably play a role. So, the authors should comment on the question what can be learned by their classification for mixed states? For instance, assume that all their classification problems for two copies were solved, what does it imply for the mixed state case? This is an essential question to judge the long-term impact of this work.

We naturally agree with the observation that the states one has to deal with in experiments are mixed but believe that it is nevertheless important to study pure state transformations for two reasons. First, from a theoretical point of view, understanding pure state transformations is a necessary step towards the study of mixed state transformations. Second, for experimental states that are "close" enough to pure states, the pure state transformations hold up to a certain fidelity of the final states. We added a paragraph summarizing the remark of the referee, as well as the two arguments given above, in the introduction of our paper.

even if one restricts the attention to pure states: What are the interesting problems for further studies and what are the long term applications? The conclusion does not answer this.

We agree that the impact of our results on further studies could be made clearer. We therefore modified our conclusions to stress that our results indicate that the structure of multi-state LOCC transformations is so involved (e.g. with the possibility of changing the SLOCC class of some of the states) that a complete characterization is unlikely. We also added that further studies should therefore focus on the asymptotic case, or on physically relevant sets of states (for which the multi-state LOCC structure might be simpler).

the readability would be improved if Figure 1 appears already in the introduction.

We followed this suggestion and moved Figure 1 to the introduction.

in the introduction, the authors write that "Therefore, all entangled bipartite states form a single SLOCC equivalence class". This seems not correct to me, as entangled states with a different Schmidt rank are not equivalent under SLOCC. The authors should be more precise here, and also later (e.g. in the first sentence of Section V).

We agree that this statement should be more precisely stated. We added in the introduction that we consider states with the same local ranks. We believe that it is not necessary to emphasize this again in section V, as the statement there explicitly refers to fully-entangled states, i.e. states for which all the single-party reduced density matrices have full rank.

at first sight, for the problem stated in Eq. 56 seems there is a trivial example, namely the following: If one takes as \mu two copies of a product state, and as \nu two copies of a Bell state, then one can argue as in Figure 2, and one finds an instance where the conditions of Eq. 56 hold. Is this reasoning correct? If yes, then maybe the authors should not claim that situations as in Eq. 56 are "surprising".

We would like to mention that the suggested transformation does not satisfy our assumptions. Indeed, for the bipartite transformations, we explicitly require that the dimensions of the initial states should be preserved through the transformation in order to avoid trivial transformations (see the first paragraph of Section V, with the example of the trivial $|\phi^+\rangle$ states transformation, which is similar to that suggested by the referee). More generally, we agree that it is not surprising to find sub-SWAP transformations (i.e. transformations as illustrated in Figure 2) also in the bipartite case. We nevertheless find more surprising that when the two initial states have coprime dimensions (in which case no sub-SWAP can be performed), many non-trivial solutions requiring various types of algebraic constraints on the Schmidt coefficients can still be found.

the permutations studied in the bipartite case (e.g., Figure 3) seem to be directly related to permutations that have been characterized in the context of high-dimensional entanglement (see Eq. 8 in Kraft et al., PRL 120, 060502 (2018)). Can the authors comment on that?

We thank the referee for pointing out this reference, where the authors study the decomposability of high-dimensional states into tensor products, as it is indeed related to the problem that we study in Section V. However, to characterize multi-state LU transformations of bipartite states, we have to look for all possible decompositions of a decomposable state and thus cannot exploit the algorithm proposed in [Kraft et al., PRL 120, 060502 (2018)], which decides whether a state is decomposable or not. We added a reference to this paper, as a related work, but also stressed the difference with our problem as we did above.

We hope this reply provides the requested clarification and would like to thank the referee once again, as we think their comments helped improving our paper.

Sincerely,

The authors