Schmidt number of bipartite and multipartite states under local projections

Chen, Lin; Yu, Yi; Tang, Wai-Shing

doi:10.1007/s11128-016-1501-y

Cited by 16 publications

(16 citation statements)

References 31 publications

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“…Theorem A.2 (Block structure from Schmidt number [42]). For d 1 ≤ d 2 consider a matrix X ∈ (M d 1 ⊗ M d 2 ) + written as…”

Section: Discussionmentioning

confidence: 99%

“…To make our article selfcontained, we will review here some results introduced in [42] and [22] to upper bound the Schmidt number of bipartite quantum states. These results are based on a technique (called Choi decomposition) from [21] allowing to decompose a k-positive map for k ≥ 2 into the sum of a completely positive map and a (k − 1)-positive map with reduced input dimension.…”

Section: Appendix A: Schmidt Number Bounds From Block Structurementioning

confidence: 99%

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When Do Composed Maps Become Entanglement Breaking?

Christandl

Müller-Hermes

Wolf

2019

Ann. Henri Poincaré

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We will call a completely positive map a quantum channel if it preserves the trace, i.e. if Tr [T (X)] = Tr [X] for any X ∈ M d 1 . While quantum channels represent general physical processes in the context of quantum information theory, entanglement breaking quantum channels represent such processes that cannot be used to distribute entanglement, and they are useless for any non-classical communication task [1]. Completely copositive quantum channels represent physical processes which are too noisy to be used for some information processing tasks (e.g. quantum communication [2]). However, unlike entanglement breaking channels they can sometimes be used for certain non-classical information processing tasks (e.g. private communication [3]). * Electronic address: christandl@math.ku.dk † Electronic address: muellerh@posteo.net, muellerh@math.ku.dk ‡ Electronic address: m.wolf@tum.de

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“…Theorem A.2 (Block structure from Schmidt number [42]). For d 1 ≤ d 2 consider a matrix X ∈ (M d 1 ⊗ M d 2 ) + written as…”

Section: Discussionmentioning

confidence: 99%

Section: Appendix A: Schmidt Number Bounds From Block Structurementioning

confidence: 99%

When Do Composed Maps Become Entanglement Breaking?

Christandl

Müller-Hermes

Wolf

2019

Ann. Henri Poincaré

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“…In this section we will answer the following question, recently raised in the literature [18,Conjecture 36]: Are there PPT states ρ with an arbitrarily large difference SN(ρ)− SN(ρ Γ )? With the following theorem we demonstrate that this difference can be arbitrarily large as the local dimension grows.…”

Section: B Large Variation Of Schmidt Number Under Partial Transposimentioning

confidence: 99%

“…This gives rise to a natural question: Can high-dimensional entanglement occur at all in systems that are noisy enough to be PPT? By letting the local dimensions grow, examples of PPT states with increasingly high Schmidt number have been constructed in [18]. However, their Schmidt number scales only logarithmically in the local dimension.…”

Section: Introductionmentioning

confidence: 99%

High-Dimensional Entanglement in States with Positive Partial Transposition

et al. 2018

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Genuine high-dimensional entanglement, i.e. the property of having a high Schmidt number, constitutes a resource in quantum communication, overcoming limitations of low-dimensional systems. States with a positive partial transpose (PPT), on the other hand, are generally considered weakly entangled, as they can never be distilled into pure entangled states. This naturally raises the question, whether high Schmidt numbers are possible for PPT states. Volume estimates suggest that optimal, i.e. linear, scaling in local dimension should be possible, albeit without providing an insight into the possible slope. We provide the first explicit construction of a family of PPT states that achieves linear scaling in local dimension and we prove that random PPT states typically share this feature. Our construction also allows us to answer a recent question by Chen et al. on the existence of PPT states whose Schmidt number increases by an arbitrarily large amount upon partial transposition. Finally, we link the Schmidt number to entangled sub-block matrices of a quantum state. We use this connection to prove that quantum states invariant under partial transposition on the smaller of their two subsystems cannot have maximal Schmidt number. This generalizes a well-known result by Kraus et al. We also show that the Schmidt number of absolutely PPT states cannot be maximal, contributing to an open problem in entanglement theory.

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“…Refs. [12,[21][22][23][24][25]), however the question whether PPT states can be genuinely highdimensionally entangled have been investigated only recently [26][27][28]. In particular, Huber et al [28] found that for a special class of (d × d)-dimensional PPT entangled states the Schmidt number scales as d/4.…”

Section: Introductionmentioning

confidence: 99%

Class of genuinely high-dimensionally-entangled states with a positive partial transpose

Pál

Vértesi

2019

Phys. Rev. A

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Entangled states with a positive partial transpose (so-called PPT states) are central to many interesting problems in quantum theory. On one hand, they are considered to be weakly entangled, since no pure state entanglement can be distilled from them. On the other hand, it has been shown recently that some of these PPT states exhibit genuinely high-dimensional entanglement, i.e. they have a high Schmidt number. Here we investigate d × d dimensional PPT states for d ≥ 4 discussed recently by Sindici and Piani, and by generalizing their methods to the calculation of Schmidt numbers we show that a linear d/2 scaling of its Schmidt number in the local dimension d can be attained.

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Schmidt number of bipartite and multipartite states under local projections

Cited by 16 publications

References 31 publications

When Do Composed Maps Become Entanglement Breaking?

When Do Composed Maps Become Entanglement Breaking?

High-Dimensional Entanglement in States with Positive Partial Transposition

Class of genuinely high-dimensionally-entangled states with a positive partial transpose

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