Entangled subspaces and generic local state discrimination with pre-shared entanglement

Lovitz, Benjamin; Johnston, Nathaniel

doi:10.22331/q-2022-07-07-760

Cited by 4 publications

(2 citation statements)

References 35 publications

(67 reference statements)

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“…Although, it was not mentioned directly, the transpose of these vectors are nothing but the rows of the well known Vandermonde matrix (see appendix B). Following this work the usage of Vandermonde matrices in the context of entangled subspaces has gained a lot of interest [31][32][33]. All such results motivated us to search for constructing new and distinct CESs subspaces.…”

Section: Constructing Completely Entangled Subspaces Using Known Matr...mentioning

confidence: 99%

Completely entangled subspaces from Moore-like matrices

Nawareg

2023

Phys. Scr.

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Construction of completely entangled subspaces (CES) has gained a considerable attention, recently. These subspaces which contain only entangled states are of great importance for entanglement theory and also provide a valuable resource for quantum information processing tasks. The results of [Proc. Math. Sci. 114, 365 (2004)] and in particular using the properties of certain matrix, namely Vandermonde matrix, to build CES motivated us to search for new distinct CES’s. Mainly, the stimulating question of whether there are other matrices that can lead to building CESs emerged. In the current paper we give an affirmative answer to this question by providing a method for constructing CESs using the properties of Moore-like matrices. In addition, we give few examples for the proposed subspaces in case of 3-qubit and 2-qutrit systems. Then a comparison between the resulted subspaces and those constructed from Vandermonde matrix has been given for the systems understudy. The results shows that the two methods give identically the same subspaces in case of multiqubit systems. However, for multipartite systems with local dimensions d ≥ 3 the two methods gave unequivalent CES subspaces. Interestingly, the properties of the proposed Moore-like matrices provided a far rich way for constructing CES subspaces. It leads to generating as many distinct CES’s as we want for each multipartite quantum system. This is in contrary to Vandermonde-based method which can give only one CES per system. In addition, the basis for each of the given examples has been obtained in a simple form. Moreover, we evaluated the entanglement of uniformly mixed states over the obtained subspaces in terms of concurrence and geometric measure of entanglement. Since different parameters of a Moore-like matrix lead to distinct CESs for the same system, the realized results can open the door for more investigations and/or applications.

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Section: Constructing Completely Entangled Subspaces Using Known Matr...mentioning

confidence: 99%

Completely entangled subspaces from Moore-like matrices

Nawareg

2023

Phys. Scr.

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“…A completely entangled subspace in one containing no product states [6], while a genuinely entangled subspace is one containing no states that are product across any bipartition (genuine entanglement is a stricter requirement than complete entanglement) [12,13]. Completely entangled subspaces are useful for locally discriminating pure quantum states [14,15], while genuinely entangled subspaces have been shown to have applications in quantum cryptography [16].…”

Section: Introductionmentioning

confidence: 99%

A Complete Hierarchy of Linear Systems for Certifying Quantum Entanglement of Subspaces

Johnston¹,

Lovitz²,

Vijayaraghavan³

2022

Preprint

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We introduce a hierarchy of linear systems for showing that a given subspace of pure quantum states is entangled (i.e., contains no product states). This hierarchy outperforms known methods already at the first level, and it is complete in the sense that every entangled subspace is shown to be so at some finite level of the hierarchy. It generalizes straightforwardly to the case of higher Schmidt rank, as well as the multipartite cases of completely and genuinely entangled subspaces. These hierarchies work extremely well in practice even in very large quantum systems, as they can be implemented via elementary linear algebra techniques rather than the semidefinite programming techniques that are required by previously-known hierarchies.

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Multiparticle singlet states cannot be maximally entangled for the bipartitions

Bernards,

Gühne

2024

Journal of Mathematical Physics

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One way to explore multiparticle entanglement is to ask for maximal entanglement with respect to different bipartitions, leading to the notion of absolutely maximally entangled states or perfect tensors. A different path uses unitary invariance and symmetries, resulting in the concept of multiparticle singlet states. We show that these two concepts are incompatible in the sense that the space of pure multiparticle singlet states does not contain any state for which all partitions of two particles vs the rest are maximally entangled. This puts restrictions on the construction of quantum codes and contributes to discussions in the context of the anti-de Sitter/conformal field theory correspondence and quantum gravity.

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Entangled subspaces and generic local state discrimination with pre-shared entanglement

Cited by 4 publications

References 35 publications

Completely entangled subspaces from Moore-like matrices

Completely entangled subspaces from Moore-like matrices

A Complete Hierarchy of Linear Systems for Certifying Quantum Entanglement of Subspaces

Multiparticle singlet states cannot be maximally entangled for the bipartitions

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