SICs: Some Explanations

Bengtsson, Ingemar

doi:10.1007/s10701-020-00341-9

Cited by 13 publications

(8 citation statements)

References 27 publications

(49 reference statements)

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“…5 we conclude that Tr(π 0 ⊗ ρ ′ 0 + π ′ 0 ⊗ ρ 0 ) = 0 and for any a = (0, 0):Tr (π 0 ⊗ ρ ′ 0 + π ′ 0 ⊗ ρ 0 )W † a ⊗ W a = 1. (4.11)But if a 1 = 0 and a 2 = 0 then for a = (a 1 , a 2 ) we have Tr(π 0 W a ) = 0|W a |0 = 0 and Tr(ρ0 W a ) = 0|F † W a F |0 = 0|W (a 2 ,−a 1 ) |0 = 0, hence Tr (π 0 ⊗ ρ ′ 0 + π ′ 0 ⊗ ρ 0 )W † a ⊗ W a = 0a contradiction.You may ask what WH covariant solutions at 0 evade this proof.…”

mentioning

confidence: 74%

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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mentioning

confidence: 74%

On the continuous Zauner conjecture

Yakymenko

2021

Preprint

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“…⊗ 1 d we obtain a set of (d − 2) 2 equiangular tight frames, all of them spanning some d(d − 1)/2 dimensional subspace. We can regard the resulting set of subspaces as so many points in a Grassmannian, and then it can be shown that these points are equidistant with respect to the natural metric on the Grassmannian [23].…”

Section: Aligned Sics and Proto-sicsmentioning

confidence: 99%

Dimension towers of SICs. II. Some constructions

Bengtsson,

Srivastava

2022

Preprint

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A SIC is a maximal equiangular tight frame in a finite dimensional Hilbert space. Given a SIC in dimension d, there is good evidence that there always exists an aligned SIC in dimension d(d − 2), having predictable symmetries and smaller equiangular tight frames embedded in them. We provide a recipe for how to calculate sets of vectors in dimension d(d − 2) that share these properties. They consist of maximally entangled vectors in certain subspaces defined by the numbers entering the d dimensional SIC. However, the construction contains free parameters and we have not proven that they can always be chosen so that one of these sets of vectors is a SIC. We give some worked examples that, we hope, may suggest to the reader how our construction can be improved. For simplicity we restrict ourselves to the case of odd dimensions.

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“…This scheme, often called SIC in the literature, is proven to be optimal for quantum state tomography [8]. Such a particular generalized measurements known for a fast-growing number of dimensions [13,14,15,16,17], but its existence for an arbitrary N has not yet been established [18,19].…”

Section: Introductionmentioning

confidence: 99%

“…Zauner conjectured [24,25] that such ensembles exist for any N . Analytic constructions are known [17,14] for all dimensions N ≤ 53, while numerical solutions were obtained [13,18,26] for dimensions N ≤ 193, with some additional solutions known for certain dimensions [15,16,19] up to N = 2208.…”

Section: Introductionmentioning

confidence: 99%

Bipartite quantum measurements with optimal single-sided distinguishability

Czartowski

Życzkowski

2021

Quantum

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We analyse orthogonal bases in a composite N×N Hilbert space describing a bipartite quantum system and look for a basis with optimal single-sided mutual state distinguishability. This condition implies that in each subsystem the N2 reduced states form a regular simplex of a maximal edge length, defined with respect to the trace distance. In the case N=2 of a two-qubit system our solution coincides with the elegant joint measurement introduced by Gisin. We derive explicit expressions of an analogous constellation for N=3 and provide a general construction of N2 states forming such an optimal basis in HN⊗HN. Our construction is valid for all dimensions for which a symmetric informationally complete (SIC) generalized measurement is known. Furthermore, we show that the one-party measurement that distinguishes the states of an optimal basis of the composite system leads to a local quantum state tomography with a linear reconstruction formula. Finally, we test the introduced tomographical scheme on a complete set of three mutually unbiased bases for a single qubit using two different IBM machines.

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SICs: Some Explanations

Cited by 13 publications

References 27 publications

On the continuous Zauner conjecture

On the continuous Zauner conjecture

Dimension towers of SICs. II. Some constructions

Bipartite quantum measurements with optimal single-sided distinguishability

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