Introduction to Finite Frame Theory

Casazza, Peter G.; Kutyniok, Gitta; Philipp, Friedrich

doi:10.1007/978-0-8176-8373-3_1

Cited by 88 publications

(86 citation statements)

References 118 publications

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“…, and so a normalized rank-1 POVM can be also defined as a (multi-)set of points in P (H) that constitutes a uniform (or normalized) tight frame in P (H) [14,26,47], that is, an ensemble that fulfills k j=1 tr σ j ρ = k/d for every ρ ∈ P (H). In this case, we shall say that σ j ( j = 1, .…”

Section: Quantum States and Povmsmentioning

confidence: 99%

Highly symmetric POVMs and their informational power

Słomczyński

Szymusiak

2015

Quantum Inf Process

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We discuss the dependence of the Shannon entropy of normalized finite rank-1 POVMs on the choice of the input state, looking for the states that minimize this quantity. To distinguish the class of measurements where the problem can be solved analytically, we introduce the notion of highly symmetric POVMs and classify them in dimension 2 (for qubits). In this case, we prove that the entropy is minimal, and hence, the relative entropy (informational power) is maximal, if and only if the input state is orthogonal to one of the states constituting a POVM. The method used in the proof, employing the Michel theory of critical points for group action, the Hermite interpolation, and the structure of invariant polynomials for unitary-antiunitary groups, can also be applied in higher dimensions and for other entropy-like functions. The links between entropy minimization and entropic uncertainty relations, the Wehrl entropy, and the quantum dynamical entropy are described.

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Section: Quantum States and Povmsmentioning

confidence: 99%

Highly symmetric POVMs and their informational power

Słomczyński

Szymusiak

2015

Quantum Inf Process

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“…However, if we allow more freedom in the sense of choosing (φ i ) i∈I to form a frame -a redundant, yet stable system (see Subsection 3.3) -the sequences (c i ( f )) i∈I might be chosen significantly sparser for each f ∈ C . Thus, methodologies from frame theory will come into play, see Subsection 3.3 and [5,7].…”

Section: The Role Of Applied Harmonic Analysismentioning

confidence: 99%

Introduction to Shearlets

Kutyniok

Labate

2012

Applied and Numerical Harmonic Analysis

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Shearlets emerged in recent years among the most successful frameworks for the efficient representation of multidimensional data. Indeed, after it was recognized that traditional multiscale methods are not very efficient at capturing edges and other anisotropic features which frequently dominate multidimensional phenomena, several methods were introduced to overcome their limitations. The shearlet representation stands out since it offers a unique combinations of some highly desirable properties: it has a single or finite set of generating functions, it provides optimally sparse representations for a large class of multidimensional data, it is possible to use compactly supported analyzing functions, it has fast algorithmic implementations and it allows a unified treatment of the continuum and digital realms. In this chapter, we present a self-contained overview of the main results concerning the theory and applications of shearlets.

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“…(20) with A = B = 2 N and M = 4 N . According to proposition 22 in [32], a frame of a Hilbert space H with bounds A, B that is orthogonally projected to a subspace P H is a frame of P H with the same bounds A, B. Therefore we have as a corollary of Theorem 1 that the set of covariant matrices S µ1µ2...µN forms a 2 N -tight frame for L(H S ).…”

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confidence: 89%

“…In order to fully exploit the consequences of this fact, we need some basic notions of frame theory [32].…”

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confidence: 99%

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Tensor Representation of Spin States

et al. 2015

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We propose a generalization of the Bloch sphere representation for arbitrary spin states. It provides a compact and elegant representation of spin density matrices in terms of tensors that share the most important properties of Bloch vectors. Our representation, based on covariant matrices introduced by Weinberg in the context of quantum field theory, allows for a simple parametrization of coherent spin states, and a straightforward transformation of density matrices under local unitary and partial tracing operations. It enables us to provide a criterion for anticoherence, relevant in a broader context such as quantum polarization of light. The concept of spin is ubiquitous in quantum theory and all related fields of research, such as solidstate physics, molecular, atomic, nuclear or high-energy physics [1][2][3][4][5]. It has profound implications for the structure of matter as a consequence of the celebrated spinstatistics theorem [6]. The spin of a quantum system, be it an electron, a nucleus or an atom, has also been proven to be a key resource for many applications such as in spintronics [7], quantum information theory [8] or nuclear magnetic resonance [9]. Simple geometrical representations of spin states [10] allow one to develop physical insight regarding their general properties and evolution. Particularly well studied is the case of a single twolevel system, formally equivalent to a spin-1/2. In this case, the geometric representation is particularly simple. Indeed, the density matrix can be expressed in a basis formed of Pauli matrices and the identity matrix, leading to a parametrization in terms of a vector in R 3 . Pure states correspond to points on a unit sphere, the so-called Bloch sphere, and mixed states fill the inside of the sphere, the "Bloch ball". The simplicity of this representation help visualize the action and geometry of all possible spin-1/2 quantum channels [11]. For arbitrary pure spin states, another nice geometrical representation has been developed by Majorana in which a spin-j state is visualized as 2j points on the Bloch sphere [12]. This so-called Majorana or stellar representation has been exploited in various contexts (see, e. g., [11,[13][14][15][16][17]), but cannot be generalized to mixed spin states.Given the importance of geometrical representations, there have been numerous attempts to extend the previous representations to arbitrary mixed states. The former rely on a variety of sophisticated mathematical concepts such as su(N )-algebra generators [10,18,19], polarization operator basis [20][21][22] [26]. In the present Letter we propose an elegant generalisation to arbitrary spin-j of the spin-1/2 Bloch sphere representation based on matrices introduced by Weinberg in the context of relativistic quantum field theory [27]. The main result of the paper is theorem 2, which allows us to express any spin-j density matrix as a linear combination of matrices with convenient properties. The remarkable features of our representation are especially reflected in the simple coo...

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Introduction to Finite Frame Theory

Cited by 88 publications

References 118 publications

Highly symmetric POVMs and their informational power

Highly symmetric POVMs and their informational power

Introduction to Shearlets

Tensor Representation of Spin States

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