Entanglement breaking rank and the existence of SIC POVMs

Pandey, Satish K.; Paulsen, Vern I.; Prakash, J.; Rahaman, Mizanur

doi:10.1063/1.5045184

Cited by 12 publications

(12 citation statements)

References 23 publications

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“…In a recent paper by S.Pandey, V.Paulsen, J.Prakash, and M.Rahaman [15], the authors studied the entanglement breaking quantum channels Φ t :…”

Section: Introductionmentioning

confidence: 99%

“…One can readily see that for t = 1 d+1 the two frames should coincide (up to phase factors) and satisfy SIC relations. Moreover, the results of [15] imply that there exists a family of MUFs which is continuous over t on [0, 1 d+1 ] in dimensions 2 and 3 (with a nuance at t = 0). Therefore, the PPPR conjecture may be seen as a continuous extension of Zauner's conjecture.…”

Section: Introductionmentioning

confidence: 99%

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On the continuous Zauner conjecture

Yakymenko

2021

Preprint

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In a recent paper by S.Pandey, V.Paulsen, J.Prakash, and M.Rahaman, the authors studied the entanglement breaking quantum channels Φt :They proved that Zauner's conjecture is equivalent to the statement that entanglement breaking rank of Φ 1 d+1 is d 2 . The authors made the extended conjecture that ebr(Φt) = d 2 for every t ∈ [0, 1 d+1 ] and proved it in dimensions 2 and 3. In this paper we prove that for anythe equality ebr(Φt) = d 2 is equivalent to the existence of a pair of informationally complete unit norm tight frames {|xi } d 2 i=1 , {|yi } d 2 i=1 in C d which are mutually unbiased in a certain sense. That is, for any i = j it holds that | xi|yj | 2 = 1−t d and | xi|yi | 2 = t(d 2 −1)+1 d (also it follows that | xi|xj yi|yj | = |t|). Though, our numerical searches for solutions were not successful in dimensions 4 and 5 for values of t other than 0 or 1 d+1 .

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“…In a recent paper by S.Pandey, V.Paulsen, J.Prakash, and M.Rahaman [15], the authors studied the entanglement breaking quantum channels Φ t :…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On the continuous Zauner conjecture

Yakymenko

2021

Preprint

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“…Pandey et al [44] recently proposed a new approach to Zauner's conjecture. They identify an explicit quantum channel :…”

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confidence: 99%

“…In this paper, we study tight projective 2-designs in three different settings. In Section 2, we consider the complex setting, specifically, the new quantitative approach of Pandey et al [44].…”

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confidence: 99%

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A note on tight projective 2‐designs

Iverson

King

Mixon

2021

J of Combinatorial Designs

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We study tight projective 2-designs in three different settings. In the complex setting, Zauner's conjecture predicts the existence of a tight projective 2-design in every dimension. Pandey, Paulsen, Prakash, and Rahaman recently proposed an approach to make quantitative progress on this conjecture in terms of the entanglement breaking rank of a certain quantum channel. We show that this quantity is equal to the size of the smallest weighted projective 2-design. Next, in the finite field setting, we introduce a notion of projective 2-designs, we characterize when such projective 2-designs are tight, and we provide a construction of such objects. Finally, in the quaternionic setting, we show that every tight projective 2-design for d determines an equi-isoclinic tight fusion frame of d d(2 − 1) subspacesThe vertices of the tetrahedron provide a famously nice arrangement of points on the unit sphere in 3 . This highly symmetric configuration was constructed thousands of years ago in Euclid's Elements, and it has since emerged as an optimal arrangement for several fundamental problems in metric geometry [3,18]. For example, it solves the n = 4 case of Tammes's inimical dictators problem [19], which asks to maximize the minimum distance between n dictators on the sphere. It also solves the n = 4 case of Thomson's energy minimization problem [50], and more generally, it minimizes every completely monotonic potential [12], which explains the tetrahedral shape exhibited by the methane molecule [30].

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Optimal Quantum Measurements

Stacey

2021

A First Course in the Sporadic SICs

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Entanglement breaking rank and the existence of SIC POVMs

Cited by 12 publications

References 23 publications

On the continuous Zauner conjecture

On the continuous Zauner conjecture

A note on tight projective 2‐designs

Optimal Quantum Measurements

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