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Local Transformations of Multiple Multipartite States
by Antoine Neven, David Gunn, Martin Hebenstreit, Barbara Kraus
This is not the current version.
|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|
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.
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Anonymous Report 1 on 2021-1-24 (Invited 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?