Distinguishability of complete and unextendible product bases

Rinaldis, S. De

doi:10.1103/physreva.70.022309

Cited by 105 publications

(77 citation statements)

References 17 publications

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“…Recently, there has been a renewed wave of interest in LOCC alongside new discoveries concerning asymptotic resources in LOCC processing [Rin04,Chi11,KKB11,CLMO12]. It has now been shown that when an unbounded number of communication rounds are allowed, or when a particular task needs only to be accomplished with an arbitrarily small failure rate (but not perfectly), more can be accomplished than in the setting of finite rounds and perfect success rates.…”

Section: Introductionmentioning

confidence: 99%

Everything You Always Wanted to Know About LOCC (But Were Afraid to Ask)

Chitambar

Leung

Mančinska

et al. 2014

Commun. Math. Phys.

357

371

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In this paper we study the subset of generalized quantum measurements on finite dimensional systems known as local operations and classical communication (LOCC). While LOCC emerges as the natural class of operations in many important quantum information tasks, its mathematical structure is complex and difficult to characterize. Here we provide a precise description of LOCC and related operational classes in terms of quantum instruments. Our formalism captures both finite round protocols as well as those that utilize an unbounded number of communication rounds. While the set of LOCC is not topologically closed, we show that finite round LOCC constitutes a compact subset of quantum operations. Additionally we show the existence of an open ball around the completely depolarizing map that consists entirely of LOCC implementable maps. Finally, we demonstrate a two-qubit map whose action can be approached arbitrarily close using LOCC, but nevertheless cannot be implemented perfectly.

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Section: Introductionmentioning

confidence: 99%

Everything You Always Wanted to Know About LOCC (But Were Afraid to Ask)

Chitambar

Leung

Mančinska

et al. 2014

Commun. Math. Phys.

357

371

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“…The attempt at a characterization of GESs made by us in [8] was qualitative, in a sense that we have only considered the problem of their general constructions in setups with an arbitrary number of parties holding subsystems of arbitrary local dimensions (see [19] for recent advances). This has been linked with the notion of the unextendible product bases [20], another very powerful tool with diverse applications (see, e.g., [21][22][23]). Clearly, however, the quantitative description of GESs (or, more generally, any subspaces) is also vital, as it provides a means of comparing them and potentially deciding on their usefulness in certain tasks.…”

Section: Introductionmentioning

confidence: 99%

Entanglement of genuinely entangled subspaces and states: Exact, approximate, and numerical results

Demianowicz

Augusiak

2019

Phys. Rev. A

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Genuinely entangled subspaces (GESs) are those subspaces of multipartite Hilbert spaces that consist only of genuinely multiparty entangled (GME) pure states. They are natural generalizations of the well-known notion of completely entangled subspaces (CESs), which by definition are void of fully product vectors. Entangled subspaces are an important tool of quantum information theory as they directly lead to construtions of entangled states, since any state supported on such a subspace is automatically entangled. Moreover, they have also proven useful in the area of quantum error correction (QEC). In our recent contribution [M. Demianowicz and R. Augusiak, Phys. Rev. A 98, 012313 (2018)], we have studied the notion of a GES qualitatively in relation to socalled nonorthogonal unextendible product bases and provided a few novel constructions of such subspaces. The main aim of the present work is to perform a quantitative study of the entanglement properties of GESs. First, we show how one can attempt to compute analytically the subspace entanglement, defined as the entanglement of the least entangled vector from the subspace, of a GES and illustrate our method applying it to a new class of GESs. Second, we show that certain semi-definite programming (SDP) relaxations can be exploited to estimate the entanglement of a GES and apply this observation to few classes of GESs revealing that in many cases the method provides the exact results. Finally, we study the entanglement of certain states supported on GESs, which is compared to the obtained values of the entanglement of the corresponing subspaces, and find the white noise robustness of several GESs. In our study we use the (generalized) geometric measure as the quantifier of entanglement.

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“…It is well known that the set of UPB constitutes a special class of locally indistinguishable product states [16,30]. So it is interesting to ask whether there are some other classes of locally indistinguishable orthogonal product states except the UPB.…”

Section: Introductionmentioning

confidence: 99%

“…Some of them considered the set with maximally entangled states [5][6][7][8][9][10][11][12][13][14], while the others aimed at the set with product states [2,[15][16][17][18][19][20][21][22][23][24][25][26]. Both of these researches can lead us a better understanding the limitation of the local operations and classical communication.…”

Section: Introductionmentioning

confidence: 99%

Nonlocality of orthogonal product-basis quantum states

et al. 2015

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We study the local indistinguishability of mutually orthogonal product basis quantum states in the high-dimensional quantum system. In the quantum system of C d ⊗ C d , where d is odd, Zhang et al have constructed d 2 orthogonal product basis quantum states which are locally indistinguishable in [Phys. Rev. A. 90, 022313(2014)]. We find a subset contains with 6d − 9 orthogonal product states which are still locally indistinguishable. Then we generalize our method to arbitrary bipartite quantum system C m ⊗ C n . We present a small set with only 3(m + n) − 9 orthogonal product states and prove these states are LOCC indistinguishable. Even though these 3(m + n) − 9 product states are LOCC indistinguishable, they can be distinguished by separable measurements. This shows that separable operations are strictly stronger than the local operations and classical communication.

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Distinguishability of complete and unextendible product bases

Cited by 105 publications

References 17 publications

Everything You Always Wanted to Know About LOCC (But Were Afraid to Ask)

Everything You Always Wanted to Know About LOCC (But Were Afraid to Ask)

Entanglement of genuinely entangled subspaces and states: Exact, approximate, and numerical results

Nonlocality of orthogonal product-basis quantum states

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