Construction of Mercedes–Benz Frame in $${\mathbb {R}}^n$$ R n

Parvathalu, B.; Johnson, P. Sam

doi:10.1007/s40819-017-0367-8

Cited by 4 publications

(7 citation statements)

References 5 publications

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“…MB frames in ℝ d are defined in [9][10][11][12] in a natural way based on Example 1 and Example 2, where they assume that the vector (0, ⋯ , 0, 1) T is one of the vectors in the system. We drop that and define the following way.…”

Section: Mercedes-benz Frames In ℝ Dmentioning

confidence: 99%

“…The Mercedes-Benz (MB) frame in ℝ 2 provides a perfect example of an equiangular tight frame. Such frames have been generalized to higher dimensions as well [9][10][11][12]. The MB frames are a very special family of equiangular frames since the former have ⟨ v i , v j ⟩ = c whereas the latter have | ⟨ v i , v j ⟩ | = c for any pair v i , v j of frame vectors, where c is a constant.…”

Section: Introductionmentioning

confidence: 99%

“…The MB frames are a very special family of equiangular frames since the former have ⟨ v i , v j ⟩ = c whereas the latter have | ⟨ v i , v j ⟩ | = c for any pair v i , v j of frame vectors, where c is a constant. The existence of the MB frame in ℝ d is known and the constructive proof appears in [9][10][11][12]. They are uniquely determined up to projective unitary equivalence by their reduced signature matrix [15].…”

Section: Introductionmentioning

confidence: 99%

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MERCEDES-BENZ FRAMES IN $$\mathbb {R}^d$$ AS A DIRECT SUM OF A PAIR OF ORTHOGONAL TIGHT FRAMES

Bhatt

2023

J Math Sci

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The well-known Mercedes-Benz frames in ℝ d are known to exist and they have been constructed too. They are presented here as an external direct sum of a pair of orthogonal frames. A necessary and sufficient condition for a unit norm tight frame to be a Mercedes-Benz frame in ℝ d is provided in terms of a vector from the null space of its synthesis operator. Also, a necessary and sufficient condition for a unit norm system of vectors to be a Mercedes-Benz frame is provided in terms of the frame potential and a vector from the null space of the synthesis operator.

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Section: Mercedes-benz Frames In ℝ Dmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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MERCEDES-BENZ FRAMES IN $$\mathbb {R}^d$$ AS A DIRECT SUM OF A PAIR OF ORTHOGONAL TIGHT FRAMES

Bhatt

2023

J Math Sci

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“…Discriminating between quantum channels representing C 3 has some precedent in prior work: such channels are precisely those which can generate the Peres-Wootters states [38,39] or equivalently Mercedes-Benz frames [40,41] (for their threefold symmetry).…”

Section: A Cyclic Groupsmentioning

confidence: 99%

Quantum Hypothesis Testing with Group Structure

Rossi,

Chuang

2021

Preprint

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The problem of discriminating between many quantum channels with certainty is analyzed under the assumption of prior knowledge of algebraic relations among possible channels. It is shown, by explicit construction of a novel family of quantum algorithms, that when the set of possible channels faithfully represents a finite subgroup of SU(2) (e.g., Cn, D2n, A4, S4, A5) the recentlydeveloped techniques of quantum signal processing can be modified to constitute subroutines for quantum hypothesis testing. These algorithms, for group quantum hypothesis testing (G-QHT), intuitively encode discrete properties of the channel set in SU(2) and improve query complexity at least quadratically in n, the size of the channel set and group, compared to naïve repetition of binary hypothesis testing. Intriguingly, performance is completely defined by explicit group homomorphisms; these in turn inform simple constraints on polynomials embedded in unitary matrices. These constructions demonstrate a flexible technique for mapping questions in quantum inference to the well-understood subfields of functional approximation and discrete algebra. Extensions to larger groups and noisy settings are discussed, as well as paths by which improved protocols for quantum hypothesis testing against structured channel sets have application in the transmission of reference frames, proofs of security in quantum cryptography, and algorithms for property testing.

show abstract

“…Discriminating between quantum channels representing C 3 has some precedent in prior work: such channels are precisely those which can generate the Peres-Wootters states [38,39] or equivalently the Mercedes-Benz frames [40,41] (for their threefold symmetry).…”

Section: A Cyclic Groupsmentioning

confidence: 99%

Quantum hypothesis testing with group structure

Rossi

Chuang²

2021

Phys. Rev. A

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The problem of discriminating between many quantum channels with certainty is analyzed under the assumption of prior knowledge of algebraic relations among possible channels. It is shown, by explicit construction of a novel family of quantum algorithms, that when the set of possible channels faithfully represents a finite subgroup of SU(2) (e.g., C n , D 2n , A 4 , S 4 , A 5 ) the recently developed techniques of quantum signal processing can be modified to constitute subroutines for quantum hypothesis testing. These algorithms, for group quantum hypothesis testing, intuitively encode discrete properties of the channel set in SU(2) and improve query complexity at least quadratically in n, the size of the channel set and group, compared to naïve repetition of binary hypothesis testing. Intriguingly, performance is completely defined by explicit group homomorphisms; these in turn inform simple constraints on polynomials embedded in unitary matrices. These constructions demonstrate a flexible technique for mapping questions in quantum inference to the well-understood subfields of functional approximation and discrete algebra. Extensions to larger groups and noisy settings are discussed, as well as paths by which improved protocols for quantum hypothesis testing against structured channel sets have application in the transmission of reference frames, proofs of security in quantum cryptography, and algorithms for property testing.

show abstract

Construction of Mercedes–Benz Frame in $${\mathbb {R}}^n$$ R n

Cited by 4 publications

References 5 publications

MERCEDES-BENZ FRAMES IN $$\mathbb {R}^d$$ AS A DIRECT SUM OF A PAIR OF ORTHOGONAL TIGHT FRAMES

MERCEDES-BENZ FRAMES IN $$\mathbb {R}^d$$ AS A DIRECT SUM OF A PAIR OF ORTHOGONAL TIGHT FRAMES

Quantum Hypothesis Testing with Group Structure

Quantum hypothesis testing with group structure

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