Linear dependencies in Weyl–Heisenberg orbits

Dang, Hoan Bui; Blanchfield, Kate; Bengtsson, Ingemar; Appleby, D. M.

doi:10.1007/s11128-013-0609-6

Cited by 25 publications

(48 citation statements)

References 32 publications

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“…In relation to the SIC-POVM problem we will revisit the cyclic case and prove that all eigenvectors of Clifford unitaries whose (projective) order is not coprime to the dimension N, for N odd, generate spark deficient Gabor frames, extending some results in [9]. This shows that there is not in principle any relation between these two basic properties of a Gabor frame, namely equiangularity and the full spark property.…”

Section: Introductionmentioning

confidence: 83%

“…2) and M is the operator with the property Mf (g) = ω g f (g) for all g ∈ Z/NZ and f ∈ C N , and is called the Weyl-Heisenberg group of G. Sometimes [1,9,33], these representatives over the center are considered:…”

Section: 3mentioning

confidence: 99%

“…As it has already been proven by the author [21], almost all windows generate full spark Gabor frames, so the spark deficient Gabor frames are generated by exceptional vectors. When the order of the group is an odd, square-free integer, then all eigenvectors of certain unitaries belonging to the Clifford group generate spark deficient Gabor frames [9]. The motivation behind this result in [9] was to establish a connection between equiangularity of a Gabor frame (SIC-POVM existence) and full spark, if any.…”

Section: Spark Deficient Gabor Frames Over Cyclic Groupsmentioning

confidence: 99%

“…When the order of the group is an odd, square-free integer, then all eigenvectors of certain unitaries belonging to the Clifford group generate spark deficient Gabor frames [9]. The motivation behind this result in [9] was to establish a connection between equiangularity of a Gabor frame (SIC-POVM existence) and full spark, if any. In 3 dimensions, the family of SIC-POVMs generated by vectors of the form (0, 1, −e iθ ) is always spark deficient, and Lane Hughston [15] first established a connection between the linear dependencies that arise from this SIC-POVM for θ = 0 or 2π/9 and the inflection points of an elliptic curve.…”

Section: Spark Deficient Gabor Frames Over Cyclic Groupsmentioning

confidence: 99%

“…In 3 dimensions, the family of SIC-POVMs generated by vectors of the form (0, 1, −e iθ ) is always spark deficient, and Lane Hughston [15] first established a connection between the linear dependencies that arise from this SIC-POVM for θ = 0 or 2π/9 and the inflection points of an elliptic curve. In general, it was proven in [9] that when N is an odd, square-free integer divisible by 3, all eigenvectors of the Zauner unitary matrix generate spark deficient Gabor frames. Zauner's conjecture [33] states that an eigenvector of this matrix generates a SIC-POVM, i. e. a maximal equiangular tight frame.…”

Section: Spark Deficient Gabor Frames Over Cyclic Groupsmentioning

confidence: 99%

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Spark deficient Gabor frames

Malikiosis

2018

Pacific J. Math.

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Abstract. The theory of Gabor frames of functions defined on finite abelian groups was initially developed in order to better understand the properties of Gabor frames of functions defined over the reals. However, during the last twenty years the topic has acquired an interest of its own. One of the fundamental questions asked in this finite setting is the existence of full spark Gabor frames. The author proved the existence [21], as well as constructed such frames, when the underlying group is finite cyclic. In this paper, we resolve the non-cyclic case; in particular, we show that there can be no full spark Gabor frames of windows defined on finite abelian non-cyclic groups. We also prove that all eigenvectors of certain unitary matrices in the Clifford group in odd dimensions generate spark deficient Gabor frames. Finally, similarities between the uncertainty principles concerning the finite dimensional Fourier transform and the short-time Fourier transform are discussed. IntroductionThe Gabor frame of a function f ∈ L 2 (R) is the set of all time-frequency translates of f , that is, the set of all functions of the form e 2πixy f (x−t), for y, t ∈ R, and it is a fundamental concept in time-frequency analysis and frame theory [25]. The function f usually represents a signal, t the time delay, and the pointwise multiplication by e 2πixy is the frequency "shift". Through sampling and periodization [8] one passes to the finite version of a Gabor frame, namely the shift-frequency translates of a complex function defined on a finite cyclic group. Even though finite dimensional Gabor frames were studied in order to analyze the properties of continuous signals, they later developed an interest of their own.Up to multiplication by roots of unity, a finite dimensional Gabor frame is the same as a Weyl-Heisenberg orbit, and this terminology is much more prevalent in mathematical physics and quantum information theory. A conjecture by Zauner [33] states that for every dimension N there are vectors (called "fiducials") whose WH orbit is equiangular. This means that the expression | u, v | is constant for every pair of distinct vectors u, v within this orbit. This is also known as the SIC-POVM problem which has attracted a lot of attention lately due to the vast connections to scientific areas such as quantum cryptography [26], quantum tomography [27], and algebraic number theory, especially Hilbert's 12th problem for real quadratic fields [2,3,4,5]. Such a WH orbit would then produce the maximal possible number of vectors in C N that are pairwise equiangular, namely N 2 [29]. Yet another terminology that appears for this phenomenon is maximal equiangular tight frame (or maximal ETF for short) [11], which is a special case of the packing problem in the setting of projective spaces. The interest of the algebraic construction of families of ETFs has also increased due to applications to signal processing [12,16,17].A conjecture by Heil, Ramanathan, and Topiwala from 1996 [14] states that any finite set of a Gabor frame of a ...

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Section: Introductionmentioning

confidence: 83%

Section: 3mentioning

confidence: 99%

Section: Spark Deficient Gabor Frames Over Cyclic Groupsmentioning

confidence: 99%

Section: Spark Deficient Gabor Frames Over Cyclic Groupsmentioning

confidence: 99%

Section: Spark Deficient Gabor Frames Over Cyclic Groupsmentioning

confidence: 99%

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Spark deficient Gabor frames

Malikiosis

2018

Pacific J. Math.

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Spark Deficient Gabor Frame Provides A Novel Analysis Operator For Compressed Sensing

Kouni

Rauhut

2021

Communications in Computer and Information Science

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The analysis sparsity model is a very effective approach in modern Compressed Sensing applications. Specifically, redundant analysis operators can lead to fewer measurements needed for reconstruction when employing the analysis l1-minimization in Compressed Sensing. In this paper, we pick an eigenvector of the Zauner unitary matrix and -under certain assumptions on the ambient dimension-we build a spark deficient Gabor frame. The analysis operator associated with such a spark deficient Gabor frame, is a new (highly) redundant Gabor transform, which we use as a sparsifying transform in Compressed Sensing. We conduct computational experiments -on both synthetic and real-world data-solving the analysis l1-minimization problem of Compressed Sensing, with four different choices of analysis operators, including our Gabor analysis operator. The results show that our proposed redundant Gabor transform outperforms -in all cases-Gabor transforms compared to state-of-the-art window vectors of time-frequency analysis.

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Discrimination of POVMs with rank-one effects

Krawiec

Pawela

Puchała

2020

Quantum Inf Process

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The main goal of this work is to provide an insight into the problem of discrimination of positive operator-valued measures with rank-one effects. It is our intention to study multiple-shot discrimination of such measurements, that is the case when we are able to use to unknown measurement a given number of times. Furthermore, we are interested in comparing two possible discrimination schemes: the parallel and adaptive ones. To this end, we construct a pair of symmetric informationally complete positive operator-valued measures which can be perfectly discriminated in a two-shot adaptive scheme but cannot be distinguished in the parallel scheme. On top of this, we provide an explicit algorithm which allows us to find this adaptive scheme.

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Linear dependencies in Weyl–Heisenberg orbits

Cited by 25 publications

References 32 publications

Spark deficient Gabor frames

Spark deficient Gabor frames

Spark Deficient Gabor Frame Provides A Novel Analysis Operator For Compressed Sensing

Discrimination of POVMs with rank-one effects

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