Deterministic Local Conversion of Incomparable States by Collective LOCC

Abstract

Incomparability of pure bipartite entangled states under deterministic LOCC is a very strange phenomena. We find two possible ways of getting our desired pure entangled state which is incomparable with the given input state, by collective LOCC with certainty. The first one is by providing some pure entanglement through the lower dimensional maximally-entangled states or using further less amount of entanglement and the next one is by collective operation on two pairs which are individually incomparable. It is quite surprising that we are able to achieve maximally entangled states of any Schmidt rank from a finite number of 2x2 pure entangled states only by deterministic LOCC. We provide general theory for the case of 3x3 system of incomparable states by the above processes where incomparability seems to be the most hardest one.

Conditions for entanglement transformation between a class of multipartite pure states with generalized Schmidt decompositions

“…Otherwise, every party performs a bit-flip operation X on their subsystems. We would like to point out that many results valid for bipartite pure states can be directly generalized to multipartite pure states with generalized Schmidt decompositions, as these results only depend on Schmidt coefficient vectors [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]. For instance, using the very same method, we can easily show the following generalized Vidal's formula [6] for a multipartite quantum system.…”

“…In 1999 Nielsen reported a sufficient and necessary condition for the deterministic entanglement transformations between bipartite pure states [2]. Nielsen's work has been extended to the case when a deterministic transformation cannot be achieved [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]. These efforts indicate that the structure of bipartite pure entanglement has been well understood.…”

“…As another instance, one can easily see that surprising phenomena such as entanglement catalysis [5], partial recovery of entanglement [8], multiple-copy entanglement transformation [9], and mutual catalysis [10] also exist in multipartite quantum systems, and many interesting properties concerning with these phenomena are still valid for multipartite pure states with generalized Schmidt decompositions [11,12,13,14,15,16,17,18].…”

Character of superposed states under deterministic LOCC

“…In this work we want to study the effect of superposition on states that are deterministically convertible to another state under LOCC and also the states which are not convertible. Quite interestingly we find one can generate pair of comparable states [14][15][16] through the superposition of incomparable states depending upon some specific conditions. The behavior of entanglement also changes with specific conditions under such superpositions of different kind of comparable or incomparable states.…”

General classes of impossible operations through the existence of incomparable states

“…If it happens that |Ψ → |Φ and |Φ → |Ψ then we denote it as |Ψ ↔ |Φ and describe (|Ψ , |Φ ) as a pair of incomparable states[16,33]. One of the peculiar feature of such incomparable pairs is that we are unable to say that which state has a greater amount of entanglement content than the other.…”