Five open problems in quantum information

Horodecki, Paweł; Rudnicki, Łukasz; Życzkowski, Karol

doi:10.48550/arxiv.2002.03233

Cited by 20 publications

(29 citation statements)

References 69 publications

(91 reference statements)

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“…PPT entangled states that are not separable have been the first -and, to date, only [60] -examples of entangled states that are undistillable, meaning that their distillable entanglement vanishes [61][62][63]. This is not to say that they are easy to generate; in fact, they have positive entanglement cost [64], become distillable if the set of protocols available is enlarged to include all 'PPT operations' [65], and can even have an almost maximal 'Schmidt number' [66][67][68].…”

Section: Relative Entropy Of Npt Entanglementmentioning

confidence: 99%

Attainability and lower semi-continuity of the relative entropy of entanglement, and variations on the theme

Lami¹,

Shirokov²

2021

Preprint

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The relative entropy of entanglement ER is defined as the distance of a multi-partite quantum state from the set of separable states as measured by the quantum relative entropy. We show that this optimisation is always achieved, i.e. any state admits a (unique) closest separable state, even in infinite dimension; also, ER is everywhere lower semi-continuous. These results, which seem to have gone unnoticed so far, hold not only for the relative entropy of entanglement and its multi-partite generalisations, but also for many other similar resource quantifiers, such as the relative entropy of non-Gaussianity, of non-classicality, of Wigner negativity -more generally, all relative entropy distances from the sets of states with non-negative λ-quasi-probability distribution. The crucial hypothesis underpinning all these applications is the weak*-closedness of the cone generated by free states. We complement our findings by giving explicit and asymptotically tight continuity estimates for ER and closely related quantities in the presence of an energy constraint. CONTENTS I. Introduction II. Notation A. Topologies for quantum systems B. Relative entropy of resource III. Main results A. The statements B. Proofs C. On the continuity of the relative entropy of resource D. On the continuity of the minimiser IV. Applications A. Relative entropy of entanglement B. Relative entropy of multi-partite entanglement C. Relative entropy of NPT entanglement D. Relative entropy of non-classicality, Wigner non-positivity, and generalisations thereof E. Relative entropy of non-Gaussianity V. Tight uniform continuity bounds for the relative entropy of entanglement and generalisations thereof A. Energy constraints B. Uniform continuity on energy-constrained states: bipartite case C. Uniform continuity on energy-constrained states: multipartite case VI. Conclusions and outlook VII. Acknowledgements References A. On the proof of an inequality involving the multi-partite relative entropy of entanglement

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Section: Relative Entropy Of Npt Entanglementmentioning

confidence: 99%

Attainability and lower semi-continuity of the relative entropy of entanglement, and variations on the theme

Lami¹,

Shirokov²

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…For those states that are entangled, but the PPT criterion fails, Bell states cannot be distilled that are therefore called bound entangled states or PPT-entangled states. Let us remark, it has not yet been proven, if there exists non-distillable states that have some strictly negative eigenvalues after partial transpose [7]. So curiously, states exists that can be produced by mixing maximally entangled Bell states, but those states kind of absorb the entanglement in an irreversible way, i.e.…”

Section: Introductionmentioning

confidence: 99%

Free versus Bound Entanglement: Machine learning tackling a NP-hard problem

Hiesmayr

2021

Preprint

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“…Studies of discrete structures in finite-dimensional Hilbert spaces has a long history [1,2]. Such structures are interesting not only in their own rights but also due to potential application in quantum physics.…”

Section: Introductionmentioning

confidence: 99%

“…Quantum and unitary designs are now considered as a powerful tool in quantum information science [5][6][7][8]. On the other hand, the question of building such structures are often difficult to resolve [2]. For instance, the maximal number of MUBs remains unknown even for d = 6, i.e., for the smallest dimension that is not a prime power.…”

Section: Introductionmentioning

confidence: 99%

Entropic uncertainty relations from equiangular tight frames and their applications

Rastegin¹

2021

Preprint

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Finite tight frames are interesting in various respects, including potential applications in quantum information science. Indeed, each complex tight frame leads to a non-orthogonal resolution of the identity in the Hilbert space. In a certain sense, equiangular tight frames are very similar to the maximal sets that provide symmetric informationally complete measurements. Hence, applications of equiangular tight frames in quantum physics deserve more attention than they have obtained. We derive entropic uncertainty relations for a quantum measurement built of the states of an equiangular tight frame. First, the corresponding index of coincidence is estimated from above, whence desired results follow. State-dependent and state-independent formulations are both presented. We also discuss applications of the corresponding measurements to detect entanglement and steerability.

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Five open problems in quantum information

Cited by 20 publications

References 69 publications

Attainability and lower semi-continuity of the relative entropy of entanglement, and variations on the theme

Attainability and lower semi-continuity of the relative entropy of entanglement, and variations on the theme

Free versus Bound Entanglement: Machine learning tackling a NP-hard problem

Entropic uncertainty relations from equiangular tight frames and their applications

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