On decomposable correlation matrices

Lovitz, Benjamin

doi:10.1080/03081087.2019.1661347

Cited by 3 publications

(7 citation statements)

References 18 publications

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“…We notice that the same result has also been obtained in Ref. [4] by considering the decomposition of a Gram matrix.…”

Section: A the Definition And The Lower Bound Of The Number Of Statessupporting

confidence: 86%

“…Recently a twist on this problem was proposed by some of the authors here independently [4,5], in different mathematical approach but with similar physical meaning. The main idea consists in considering sets of quantum states.…”

Section: Introductionmentioning

confidence: 99%

“…This question was discussed in Refs. [4,5]. First, a general lower bound on the size of an AES was derived: for the case of pure states in…”

Section: Introductionmentioning

confidence: 99%

“…one needs at least max (d 1 , d 2 ) + 2 states [4,5]. Second, an explicit construction of AES featuring d 1 + d 2 states was presented [this construction describes therefore a minimal set for min(d 1 , d 2 ) = 2].…”

Section: Introductionmentioning

confidence: 99%

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Absolutely entangled sets of pure states for bipartitions and multipartitions

Jayachandran

Burchardt

et al. 2021

Phys. Rev. A

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A set of quantum states is said to be absolutely entangled when at least one state in the set remains entangled for any definition of subsystems, i.e., for any choice of the global reference frame. In this work we investigate the properties of absolutely entangled sets (AESs) of pure quantum states. For the case of a two-qubit system, we present a sufficient condition to detect an AES, and use it to construct families of N states such that N − 3 (the maximal possible number) remain entangled for any definition of subsystems. For a general bipartition d = d 1 d 2 , we prove that sets of N > (d 1 + 1)(d 2 + 1)/2 states are AESs with Haar measure 1. Then, we define AESs for multipartitions. We derive a general lower bound on the number of states in an AES for a given multipartition, and also construct explicit examples. In particular, we exhibit an AES with respect to any possible multipartitioning of the total system.

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“…We notice that the same result has also been obtained in Ref. [4] by considering the decomposition of a Gram matrix.…”

Section: A the Definition And The Lower Bound Of The Number Of Statessupporting

confidence: 86%

Section: Introductionmentioning

confidence: 99%

“…This question was discussed in Refs. [4,5]. First, a general lower bound on the size of an AES was derived: for the case of pure states in…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Absolutely entangled sets of pure states for bipartitions and multipartitions

Jayachandran

Burchardt

et al. 2021

Phys. Rev. A

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“…It would be interesting to see whether our more general results could be used to characterize preservers of tensor rank r ≥ 2. In [Lov18], the author uses Corollary 9 and Corollary 10 to study the set of decomposable correlation matrices. As one more application, we use Corollary 10 to provide a concise proof of a recent result in [BLM17] (Corollary 11).…”

Section: Introductionmentioning

confidence: 99%

Toward a generalization of Kruskal's theorem on tensor decomposition

Lovitz

2018

Preprint

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Kruskal's theorem gives sufficient conditions for a sum of product vectors in general position to constitute a unique tensor rank decomposition. We conjecture an analogous result for product vectors not necessarily in general position, and show that it would imply Kruskal's theorem as a corollary. We prove our conjecture for the case in which every subsystem has dimension two, and the case in which there are only two subsystems of arbitrary dimension. We then use Kruskal's theorem to prove a family of statements on product vectors in general position that contains recent results in [HK15], and propose an analogous family of statements for product vectors not necessarily in general position.

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A generalization of Kruskal’s theorem on tensor decomposition

Lovitz¹,

Petrov

2023

Forum of Mathematics, Sigma

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Kruskal’s theorem states that a sum of product tensors constitutes a unique tensor rank decomposition if the so-called k-ranks of the product tensors are large. We prove a ‘splitting theorem’ for sets of product tensors, in which the k-rank condition of Kruskal’s theorem is weakened to the standard notion of rank, and the conclusion of uniqueness is relaxed to the statement that the set of product tensors splits (i.e., is disconnected as a matroid). Our splitting theorem implies a generalization of Kruskal’s theorem. While several extensions of Kruskal’s theorem are already present in the literature, all of these use Kruskal’s original permutation lemma and hence still cannot certify uniqueness when the k-ranks are below a certain threshold. Our generalization uses a completely new proof technique, contains many of these extensions and can certify uniqueness below this threshold. We obtain several other useful results on tensor decompositions as consequences of our splitting theorem. We prove sharp lower bounds on tensor rank and Waring rank, which extend Sylvester’s matrix rank inequality to tensors. We also prove novel uniqueness results for nonrank tensor decompositions.

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On decomposable correlation matrices

Cited by 3 publications

References 18 publications

Absolutely entangled sets of pure states for bipartitions and multipartitions

Absolutely entangled sets of pure states for bipartitions and multipartitions

Toward a generalization of Kruskal's theorem on tensor decomposition

A generalization of Kruskal’s theorem on tensor decomposition

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