From unextendible product bases to genuinely entangled subspaces

Demianowicz, Maciej; Augusiak, Remigiusz

doi:10.1103/physreva.98.012313

Cited by 50 publications

(94 citation statements)

References 48 publications

(99 reference statements)

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“…This notion is naturally generalized to the case of GME. Formally, a subspace G of a multipartite Hilbert space is called a genuinely entangled subspace (GES) if all |ψ ∈ G are genuinely entangled [8] (see also [9,10]). A simple example of a twodimensional GES is the subspace spanned by the W state, |W = 1/ √ 2(|00 .…”

Section: Preliminariesmentioning

confidence: 99%

“…000 ), and its complementW , |W = σ ⊗n x |W [36]. Few general constructions of higher dimensional GESs have been recentely given in [8], where the notion has been linked to the notion of the unextendbile product bases. In fact, the subspaces constructed there are our testground cases in the present paper.…”

Section: Preliminariesmentioning

confidence: 99%

“…One easily sees that dim S θ 2×d N −1 = (d − 1) N −1 . Note that the maximal dimension of a GES in this setup is d N −1 − 1 [8] and it would thus be interesting to see how this subspaces could be completed to maximal GESs (but not with random vectors, which can always be done).…”

Section: Entanglement Of a Class Of Genuinelymentioning

confidence: 99%

“…As the second example, we consider the subspace introduced in Theorem 2 of Ref. [8]. Originally, it has only been defined as a subspace of (C d ) ⊗N orthogonal to a continuous set of product vectors (a non-orthogonal unextendible product basis) with no explicit basis given.…”

Section: Lower Bounds On the Entanglement Of A Subspace In Terms Omentioning

confidence: 99%

“…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]).…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

Section: Entanglement Of a Class Of Genuinelymentioning

confidence: 99%

Section: Lower Bounds On the Entanglement Of A Subspace In Terms Omentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Demianowicz

Augusiak

2019

Phys. Rev. A

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Multidisciplinary Efforts Advance Materials Research at Beihang University

Cheng

Jiang

2017

Advanced Materials

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Gaze estimation methods estimate gaze from facial appearance with a single camera. However, due to the limited view of a single camera, the captured facial appearance cannot provide complete facial information and thus complicate the gaze estimation problem. Recently, camera devices are rapidly updated. Dual cameras are affordable for users and have been integrated in many devices. This development suggests that we can further improve gaze estimation performance with dual-view gaze estimation. In this paper, we propose a dual-view gaze estimation network (DV-Gaze). DV-Gaze estimates dual-view gaze directions from a pair of images. We first propose a dual-view interactive convolution (DIC) block in DV-Gaze. DIC blocks exchange dual-view information during convolution in multiple feature scales. It fuses dual-view features along epipolar lines and compensates for the original feature with the fused feature. We further propose a dual-view transformer to estimate gaze from dual-view features. Camera poses are encoded to indicate the position information in the transformer. We also consider the geometric relation between dual-view gaze directions and propose a dual-view gaze consistency loss for DV-Gaze. DV-Gaze achieves state-of-the-art performance on ETH-XGaze and EVE datasets. Our experiments also prove the potential of dual-view gaze estimation. We release codes in https://github.com/yihuacheng/DVGaze.

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