# Local Transformations of Multiple Multipartite States

### Submission summary

 As Contributors: David Kenworthy Gunn · Antoine Neven Arxiv Link: https://arxiv.org/abs/2007.06256v1 (pdf) Date submitted: 2020-09-24 17:43 Submitted by: Gunn, David Kenworthy Submitted to: SciPost Physics Academic field: Physics Specialties: Quantum Physics Approach: Theoretical

### Abstract

Understanding multipartite entanglement is vital, as it underpins a wide range of phenomena across physics. The study of transformations of states via Local Operations assisted by Classical Communication (LOCC) allows one to quantitatively analyse entanglement, as it induces a partial order in the Hilbert space. However, it has been shown that, for systems with fixed local dimensions, this order is generically trivial, which prevents relating multipartite states to each other with respect to any entanglement measure. In order to obtain a non-trivial partial ordering, we study a physically motivated extension of LOCC: multi-state LOCC. Here, one considers simultaneous LOCC transformations acting on a finite number of entangled pure states. We study both multipartite and bipartite multi-state transformations. In the multipartite case, we demonstrate that one can change the stochastic LOCC (SLOCC) class of the individual initial states by only applying Local Unitaries (LUs). We show that, by transferring entanglement from one state to the other, one can perform state conversions not possible in the single copy case; provide examples of multipartite entanglement catalysis; and demonstrate improved probabilistic protocols. In the bipartite case, we identify numerous non-trivial LU transformations and show that the source entanglement is not additive. These results demonstrate that multi-state LOCC has a much richer landscape than single-state LOCC.

###### Current status:
Has been resubmitted

### Submission & Refereeing History

Resubmission 2007.06256v3 on 11 June 2021
Resubmission 2007.06256v2 on 15 February 2021

Submission 2007.06256v1 on 24 September 2020

## Reports on this Submission

### Report

In this paper, the authors study multi-copy transformations between pure quantum states. In detail, they consider LOCC protocols acting on several copies, and present various results (for the bipartite and multipartite case) about conditions under which states can be transformed into each other on a multi-copy level.

The paper is well written and comprehensive. At the moment, however, I am not entirely sure what long term impact can be expected from this work (see the acceptance criteria of SciPost and also below). Therefore, I would like to ask the authors to revise their paper.

I have the following remarks:

- 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.

- 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.

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

- 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".

- 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?

• validity: top
• significance: good
• originality: high
• clarity: top
• formatting: excellent
• grammar: excellent

### Author:  Antoine Neven  on 2021-02-08

(in reply to Report 1 on 2021-01-24)

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