Entanglement properties of positive operators with ranges in completely entangled subspaces

Unextendible product bases (UPBs) are interesting mathematical objects arising in composite Hilbert spaces that have found various applications in quantum information theory, for instance in a construction of bound entangled states or Bell inequalities without quantum violation. They are closely related to another important notion, completely entangled subspaces (CESs), which are those that do not contain any fully separable pure state. Among CESs one finds a class of subspaces in which all vectors are not only entangled, but are genuinely entangled. Here we explore the connection between UPBs and such genuinely entangled subspaces (GESs) and provide classes of nonorthogonal UPBs that lead to GESs for any number of parties and local dimensions. We then show how these subspaces can be immediately utilized for a simple general construction of genuinely entangled states in any such multipartite scenario.

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“…Completely entangled subspaces have been a subject of intensive studies in the literature [18,19,[36][37][38]. In particular, in Refs.…”

Section: Preliminariesmentioning

confidence: 99%

From unextendible product bases to genuinely entangled subspaces

Demianowicz

Augusiak

2018

Phys. Rev. A

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“…The study of multipartite entanglement has led to the discovery of different types of entanglement that can be shared by three or more quantum particles [18][19][20]. In turn, subspaces whose vectors show interesting entanglement properties have been investigated, such as those showing a bounded Schmidt rank [21], having a negative partial transpose [22], and being completely [23,24] or genuinly entangled [25].…”

Section: Introductionmentioning

confidence: 99%

Quantum Codes of Maximal Distance and Highly Entangled Subspaces

Huber

Grassl

2020

Quantum

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We present new bounds on the existence of general quantum maximum distance separable codes (QMDS): the length n of all QMDS codes with local dimension D and distance d≥3 is bounded by n≤D2+d−2. We obtain their weight distribution and present additional bounds that arise from Rains' shadow inequalities. Our main result can be seen as a generalization of bounds that are known for the two special cases of stabilizer QMDS codes and absolutely maximally entangled states, and confirms the quantum MDS conjecture in the special case of distance-three codes. As the existence of QMDS codes is linked to that of highly entangled subspaces (in which every vector has uniform r-body marginals) of maximal dimension, our methods directly carry over to address questions in multipartite entanglement.

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“…As we have noted before, this bound was shown to be attainable by an NPT subspace in the bipartite case (in which case the bound simplifies to [13], and partial progress on this problem was made in the multipartite case in [29]. We now solve this problem by showing that there is an NPT subspace that attains the Parthasarathy bound in the multipartite case as well.…”

Section: A Multipartite Npt Subspacementioning

confidence: 73%

The Non-m-Positive Dimension of a Positive Linear Map

Johnston

Lovitz

Puzzuoli

2019

Quantum

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We introduce a property of a matrix-valued linear map Φ that we call its "non-m-positive dimension" (or "non-mP dimension" for short), which measures how large a subspace can be if every quantum state supported on the subspace is non-positive under the action of I m ⊗ Φ. Equivalently, the non-mP dimension of Φ tells us the maximal number of negative eigenvalues that the adjoint map I m ⊗ Φ * can produce from a positive semidefinite input. We explore the basic properties of this quantity and show that it can be thought of as a measure of how good Φ is at detecting entanglement in quantum states. We derive nontrivial bounds for this quantity for some well-known positive maps of interest, including the transpose map, reduction map, Choi map, and Breuer-Hall map. We also extend some of our results to the case of higher Schmidt number as well as the multipartite case. In particular, we construct the largest possible multipartite subspace with the property that every state supported on that subspace has non-positive partial transpose across at least one bipartite cut, and we use our results to construct multipartite decomposable entanglement witnesses with the maximum number of negative eigenvalues.

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Entanglement properties of positive operators with ranges in completely entangled subspaces

Cited by 8 publications

References 27 publications

From unextendible product bases to genuinely entangled subspaces

From unextendible product bases to genuinely entangled subspaces

Quantum Codes of Maximal Distance and Highly Entangled Subspaces

The Non-m-Positive Dimension of a Positive Linear Map

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