Unextendible product bases, bound entangled states, and the range criterion

Bej, Pratapaditya; Halder, Saronath

doi:10.1016/j.physleta.2020.126992

Cited by 11 publications

(11 citation statements)

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“…UPBs are connected to bound entangled states, Bell inequalities without quantum violation, and fermionic systems [7,[25][26][27][28][29]. Some results of the existence of UPBs with the minimum size were given in [30][31][32][33], and some explicit UPBs were constructed in [6,7,[34][35][36][37][38].…”

Section: Introductionmentioning

confidence: 99%

Strongly nonlocal unextendible product bases do exist

Shi

et al. 2022

Quantum

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A set of multipartite orthogonal product states is locally irreducible, if it is not possible to eliminate one or more states from the set by orthogonality-preserving local measurements. An effective way to prove that a set is locally irreducible is to show that only trivial orthogonality-preserving local measurement can be performed to this set. In general, it is difficult to show that such an orthogonality-preserving local measurement must be trivial. In this work, we develop two basic techniques to deal with this problem. Using these techniques, we successfully show the existence of unextendible product bases (UPBs) that are locally irreducible in every bipartition in d⊗d⊗d for any d≥3, and 3⊗3⊗3 achieves the minimum dimension for the existence of such UPBs. These UPBs exhibit the phenomenon of strong quantum nonlocality without entanglement. Our result solves an open question given by Halder et al. [Phys. Rev. Lett. 122, 040403 (2019)] and Yuan et al. [Phys. Rev. A 102, 042228 (2020)]. It also sheds new light on the connections between UPBs and strong quantum nonlocality.

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Section: Introductionmentioning

confidence: 99%

Strongly nonlocal unextendible product bases do exist

Shi

et al. 2022

Quantum

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“…Refs. [9][10][11][12][13][14][15][16][17][18][19], and has been experimentally verified with photons entangled in the orbital momentum degrees of freedom [20]. Often, several weakly entangled states can be transformed via local operations and classical communication (LOCC) to a smaller set of strongly entangled states.…”

Section: Introductionmentioning

confidence: 99%

Almost complete solution for the NP-hard separability problem of Bell diagonal qutrits

Popp,

Hiesmayr

2022

Preprint

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With a probability of success of 95% we solve the separability problem for Bell diagonal qutrit states with positive partial transposition (PPT). The separability problem, i.e. distinguishing separable and entangled states, generally lacks an efficient solution due to the existence of bound entangled states. In contrast to free entangled states that can be used for entanglement distillation via local operations and classical communication, these states cannot be detected by the Peres-Horodecki criterion or PPT criterion. We analyze a large family of bipartite qutrit states that can be separable, free entangled or bound entangled. Leveraging a geometrical representation of these states in Euclidean space, novel methods are presented that allow the classification of separable and bound entangled Bell diagonal states in an efficient way. Moreover, the classification allows the precise determination of relative volumes of the classes of separable, free and bound entangled states. In detail, out of all Bell diagonal PPT states 81.0% ± 0.1% are determined to be separable while 13.9 ± 0.1% are bound entangled and only 5.1 ± 0.1% remain unclassified. Moreover, our applied criteria are compared for their effectiveness and relation as detectors of bound entanglement, which reveals that not a single criterion is capable to detect all bound entangled states.

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“…[19][20][21] has shown that it can be used in the production of bound entangled (BE) states and some special bipartite entangled states that remain positive under partial transpose (PPT). Nevertheless, most of the current efforts are devoted to the construction of 2-qubit UPBs, while little progress has been made on multi-qubit UPBs [22][23][24][25][26]. Chen et al [22] investigated the minimum size of UPB with local dimension equals 2, and analyzed the proposed sets using orthogonal graphs.…”

Section: Introductionmentioning

confidence: 99%

“…Chen et al [22] investigated the minimum size of UPB with local dimension equals 2, and analyzed the proposed sets using orthogonal graphs. Bej et al [23] proposed that a set of high-dimensional reducible unextendible product states can be obtained by adding several orthogonal product states to a set of low-dimensional unextendible product states in bipartite systems. Recently, a method to construct UPBs of different large sizes in C m ⊗C n was put forward by Shi et al [24], which uses U -tile structure.…”

Section: Introductionmentioning

confidence: 99%

Strong nonlocal sets of UPB

Che,

Dou,

Lei

et al. 2021

Preprint

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The unextendible product bases (UPBs) are interesting members from the family of orthogonal product states. In this paper, we investigate the construction of 3-qubit UPB with strong nonlocality of different sizes. First, a UPB set in C 3 ⊗ C 3 ⊗ C 3 of size 12 is presented based on the Shifts UPB, the structure of which is described by mapping the system to a 3 × 3 × 3 Rubik's Cube. After observing the orthogonal graph of each qubit, we provide a general method of constructing UPB inSecond, for the more general case where the dimensions of qubits are different, we extend the tile structure to 3-qubit system and propose a Tri-tile structure for 3-qubit UPB. Then, by means of this structure, a C 4 ⊗ C 4 ⊗ C 5 system of size 30 is obtained based on a C 3 ⊗ C 3 ⊗ C 4 system. Similarly, we generalize this approach toOur research provides a positive answer to the open questions raised in [Halder, et al., PRL, 122, 040403 (2019)], indicating that there do exist multi-qubit UPBs that can exhibit strong quantum nonlocality without entanglement.

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Unextendible product bases, bound entangled states, and the range criterion

Cited by 11 publications

References 53 publications

Strongly nonlocal unextendible product bases do exist

Strongly nonlocal unextendible product bases do exist

Almost complete solution for the NP-hard separability problem of Bell diagonal qutrits

Strong nonlocal sets of UPB

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