$$4\times 4$$ unextendible product basis and genuinely entangled space

Wang, Kai; Chen, Lin; Zhao, Lei; Guo, Yu-Min

doi:10.1007/s11128-019-2324-4

Cited by 14 publications

(14 citation statements)

References 29 publications

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“…theorem 3). Fact 4 A (3,2) corresponds to a GES of H 2 3 iff the matrices X B|AC (β) (16) and X C|AB (γ ) (18) are rank four for any β and γ , respectively.…”

Section: General Casementioning

confidence: 99%

“…Primarily, these were completely entangled subspaces (CESs), that is, subspaces only with states that are in any way entangled [13][14][15]. Recently, we have witnessed an interest in so-called genuinely entangled subspaces (GESs), i.e., subspaces composed solely of genuinely multiparty entangled (GME) states, or, in other words, void of states displaying any form of separability [16][17][18][19]. The initial interest in CESs and GESs stemmed from the observation that (mixed) states supported on them are, respectively, entangled and GME.…”

Section: Introductionmentioning

confidence: 99%

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An approach to constructing genuinely entangled subspaces of maximal dimension

Demianowicz¹,

Augusiak²

2020

Quantum Inf Process

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Genuinely entangled subspaces (GESs) are the class of completely entangled subspaces that contain only genuinely multiparty entangled states. They constitute a particularly useful notion in the theory of entanglement but also have found an application, for instance, in quantum error correction and cryptography. In a recent study (Demianowicz and Augusiak in Phys Rev A 98:012313, 2018), we have shown how GESs can be efficiently constructed in any multiparty scenario from the so-called unextendible product bases. The provided subspaces, however, are not of maximal allowable dimensions, and our aim here is to put forward an approach to building such. The method is illustrated with few examples in small systems. Connections with other mathematical problems, such as spaces of matrices of equal rank and the numerical range, are discussed. Keywords Genuinely entangled subspace • Genuinely multiparty entangled • Completely entangled subspace • Entanglement • Spaces of matrices of bounded rank Maciej Demianowicz

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“…theorem 3). Fact 4 A (3,2) corresponds to a GES of H 2 3 iff the matrices X B|AC (β) (16) and X C|AB (γ ) (18) are rank four for any β and γ , respectively.…”

Section: General Casementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

An approach to constructing genuinely entangled subspaces of maximal dimension

Demianowicz¹,

Augusiak²

2020

Quantum Inf Process

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“…∏ n i=1 d i − 4 [13]. Between the minimum and maximum size, several constructions of UPBs have been given in [6,[14][15][16][17][18][19][20][21][22][23]. In particular, for UPBs in a bipartite system, DiVincenzo et al proposed a tile structure approach and constructed UPBs of size mn − 2m + 1 in C m ⊗ C n , where 3 ⩽ m ⩽ n [14].…”

Section: Introductionmentioning

confidence: 99%

Unextendible product bases from tile structures in bipartite systems

You¹,

Wang²,

Shi³

et al. 2023

J. Phys. A: Math. Theor.

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Unextendible product basis (UPB) is one of the most fundamental concepts in the field of quantum information theory, and their constructions attract much attention. Recently, Shi \emph{et al.} [\href{https://journals.aps.org/pra/abstract/10.1103/PhysRevA.101.062329}{Phy. Rev. A \textbf{101}, 062329 (2020)}] constructed UPBs via a tile structure approach. They showed that a tile structure in an $m\times n$ grid with certain properties leads to explicit constructions of UPBs in $\mathbb{C}^{m}\otimes \mathbb{C}^{n}$. The size of the UPB is $mn-s+1$ if the tile structure contains exactly $s$ tiles. In this paper, we deeply analyze the extent of the tile structure approach, by showing that a desired tile structure in an $m\times m$ grid contains at most $\lfloor \frac{m^2}{4} \rfloor + m$ tiles, and thus the smallest size of a UPB in $\mathbb{C}^{m}\otimes \mathbb{C}^{m}$ constructed by this approach is $m^2-\lfloor \frac{m^2}{4} \rfloor - m+1$. Such tile structures and UPBs are referred to as extremal tile structures and extremal UPBs. We provide algorithms both for constructing possible candidates of extremal tile structures and checking their validity. For applications, we connect the extremal UPBs to local irreducibility and bound entangled states with large ranks.

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“…For this reason, its characterization belongs naturally to the major study subjects of quantum information science. A particular trend within this branch, gaining recently considerable attention, focuses on the study of genuinely entangled subspaces (GESs) [6,7] (see also [8]), i.e., subspaces composed only of GME states, which have been identified as objects of both theoretical and practical significance [9,10,12,11,13,14]. One of the most important research lines aims at providing their constructions.…”

Section: Introductionmentioning

confidence: 99%

Universal construction of genuinely entangled subspaces of any size

Demianowicz

2022

Quantum

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We put forward a simple construction of genuinely entangled subspaces – subspaces supporting only genuinely multipartite entangled states – of any permissible dimensionality for any number of parties and local dimensions. The method uses nonorthogonal product bases, which are built from totally nonsingular matrices with a certain structure. We give an explicit basis for the constructed subspaces. An immediate consequence of our result is the possibility of constructing in the general multiparty scenario genuinely multiparty entangled mixed states with ranks up to the maximal dimension of a genuinely entangled subspace.

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$$4\times 4$$ unextendible product basis and genuinely entangled space

Cited by 14 publications

References 29 publications

An approach to constructing genuinely entangled subspaces of maximal dimension

An approach to constructing genuinely entangled subspaces of maximal dimension

Unextendible product bases from tile structures in bipartite systems

Universal construction of genuinely entangled subspaces of any size

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