Frontiers in Physics

Unextendible Entangled Bases With a Fixed Schmidt Number Based on Generalized Weighing Matrices

Yuan-Hong Tao, Xin-Lei Yong, Ya-Ru Bai, Danni Xu, Shuhui Wu

Published Jun 8, 2022• 0 citations

Abstract

We systematically study the constructions of unextendible entangled bases with a fixed Schmidt number k (UEBk) in a bipartite system Cd⊗Cd′ . Motivated by the methods of [J. Phys. A 52 : 375,303, 2019], we construct (dd’ − v)-member UEBks in Cd⊗Cd′ by using generalized weighing matrices and thus generalize the results of [arXiv: 1909.10043, 2020]. We also present the corresponding expressions of our constructions and graphically illustrate UEB3s in C5⊗C6 and C6⊗C6 .

DOI: 10.3389/fphy.2022.896164

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Frontiers in Physics

Unextendible Entangled Bases With a Fixed Schmidt Number Based on Generalized Weighing Matrices

Yuan-Hong Tao, Xin-Lei Yong, Ya-Ru Bai, Danni Xu, Shuhui Wu

Published Jun 8, 2022• 0 citations

Publisher

Abstract

DOI: 10.3389/fphy.2022.896164

Related papers

Unextendible entangled bases with fixed Schmidt number

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The unextendible product basis is generalized to the unextendible entangled basis with any arbitrarily given Schmidt number $k$ (UEBk) for any bipartite system ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{{d}^{\ensuremath{'}}}$ $(2\ensuremath{\le}k<d\ensuremath{\le}{d}^{\ensuremath{'}})$, which can also be regarded as a generalization of the unextendible maximally entangled basis. A general way of constructing such a basis with arbitrary $d$ and ${d}^{\ensuremath{'}}$ is proposed. Consequently, it is shown that there are at least $k\ensuremath{-}r$ (here $r=d\phantom{\rule{0.16em}{0ex}}\text{mod}\phantom{\rule{0.28em}{0ex}}k$ or $r={d}^{\ensuremath{'}}\phantom{\rule{0.16em}{0ex}}\text{mod}\phantom{\rule{0.28em}{0ex}}k)$ sets of UEBks when $d$ or ${d}^{\ensuremath{'}}$ is not a multiple of $k$ , while there are at least $2(k\ensuremath{-}1)$ sets of UEBks when both $d$ and ${d}^{\ensuremath{'}}$ are multiples of $k$ .The unextendible product basis is generalized to the unextendible entangled basis with any arbitrarily given Schmidt number $k$ (UEBk) for any bipartite system ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{{d}^{\ensuremath{'}}}$ $(2\ensuremath{\le}k<d\ensuremath{\le}{d}^{\ensuremath{'}})$, which can also be regarded as a generalization of the unextendible maximally entangled basis. A general way of constructing such a basis with arbitrary $d$ and ${d}^{\ensuremath{'}}$ is proposed. Consequently, it is shown that there are at least $k\ensuremath{-}r$ (here $r=d\phantom{\rule{0.16em}{0ex}}\text{mod}\phantom{\rule{0.28em}{0ex}}k$ or $r={d}^{\ensuremath{'}}\phantom{\rule{0.16em}{0ex}}\text{mod}\phantom{\rule{0.28em}{0ex}}k)$ sets of UEBks when $d$ or ${d}^{\ensuremath{'}}$ is not a multiple of $k$, while there are at least $2(k\ensuremath{-}1)$ sets of UEBks when both $d$ and ${d}^{\ensuremath{'}}$ are multiples of $k$.

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The unextendible product basis is generalized to the unextendible entangled basis with any arbitrarily given Schmidt number

k

(UEBk) for any bipartite system ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{{d}^{\ensuremath{'}}}$ $(2\ensuremath{\le}k<d\ensuremath{\le}{d}^{\ensuremath{'}})$, which can also be regarded as a generalization of the unextendible maximally entangled basis. A general way of constructing such a basis with arbitrary

d

and ${d}^{\ensuremath{'}}$ is proposed. Consequently, it is shown that there are at least $k\ensuremath{-}r$ (here

r=d\phantom{\rule{0.16em}{0ex}}\text{mod}\phantom{\rule{0.28em}{0ex}}k

or $r={d}^{\ensuremath{'}}\phantom{\rule{0.16em}{0ex}}\text{mod}\phantom{\rule{0.28em}{0ex}}k)$ sets of UEBks when

d

or ${d}^{\ensuremath{'}}$ is not a multiple of

k

, while there are at least $2(k\ensuremath{-}1)$ sets of UEBks when both

d

and ${d}^{\ensuremath{'}}$ are multiples of

k

N

-number UMEB in ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}$, there is an $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{N}$-number, $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{N}={(qd)}^{2}\ensuremath{-}({d}^{2}\ensuremath{-}N)$, UMEB in ${\mathbb{C}}^{qd}\ensuremath{\bigotimes}{\mathbb{C}}^{qd}$ for any $q\ensuremath{\in}\mathbb{N}$. As an example, for ${\mathbb{C}}^{12n}\ensuremath{\bigotimes}{\mathbb{C}}^{12n}$ systems, we show that there are at least two sets of UMEBs which are not equivalent.