Finite Tight Frames Research Articles

Enhancing some separability criteria via equiangular tight frames

Abstract Improving the detection ability of quantum entanglement is one of the es-
sential tasks in quantum computing and quantum information. Finite tight frames
play a fundamental role in a wide variety of areas, and generally, each application
requires a specific class of frames and is closely related to quantum measurement. It
is worth noting that a maximal set of complex equiangular vectors is closely related
to a symmetric informationally complete measurement. Hence, our goal in this work
is to propose a series of separability criteria assigned to a finite tight frame and some
well-known inequalities in different quantum systems, respectively. In addition, some
tighter criteria to detect entanglement for many-body quantum states are presented in
arbitrary dimensions. Finally, the effectiveness of the proposed entanglement detec-
tion criteria is illustrated through some detailed examples.

Construction of Mercedes-Benz frames via QR factorization

Finite tight frames are used widely for many applications. An important problem is to construct finite tight frames containing specified elements as needed in real world applications. The question was first studied in details by D. Feng, L. Wang, and Y. Wang in 2007. P. G. Casazza, M. Ficus, D. G. Mixon, J. Peterson, and I. Smalyanau gave a method for constructing a tight frame containing specified elements in 2013. A simple and stable method for constructing equi-norm tight frames from the specified elements was derived by J. Ji and L. Wang in 2022. As the main result, an algorithm based on the QR factorization for completing a Mercedes-Benz frame containing specified elements is presented in this paper and two examples of Mercedes-Benz frames in R4 are constructed to illustrate our approach.

Entropic uncertainty relations from equiangular tight frames and their applications

Finite tight frames are interesting in various topics including questions of quantum information. Each complex tight frame leads to a resolution of the identity in the Hilbert space. Symmetric informationally complete measurements are a special class of equiangular tight frames. Applications of such frames in quantum physics deserve more attention than they have obtained. We derive uncertainty relations for a quantum measurement assigned to an equiangular tight frame. Main results follow from estimation of the corresponding index of coincidence. State-dependent and state-independent formulations are both addressed. Also, we discuss applications of considered measurements to detect entanglement and other correlations.

Construction of finite tight frames via singular value decomposition

Finite tight frames are used widely for many applications. An important problem is to construct finite frames containing specified elements as needed in real world applications. In this paper we provide a stable method for that purpose. We establish an identity for constructing equi-norm tight frames from the specified elements and present a method for generating equi-norm tight frames explicitly from the outcome of the singular value decomposition of the matrix of the given elements. Under certain conditions, the number of columns of the unit-norm tight frames generated by our method reaches the minimum bound, improving a result in the literature. Finally, we illustrate our method through a numerical example.

Constructing Tight Gabor Frames Using CAZAC Sequences

The construction of finite tight Gabor frames plays an important role in many applications. These applications include significant ones in signal and image processing. We explore when constant amplitude zero autocorrelation (CAZAC) sequences can be used to generate tight Gabor frames. The main theorem uses Janssen’s representation and the zeros of the discrete periodic ambiguity function to give necessary and sufficient conditions for determining whether any Gabor frame is tight. The relevance of the theorem depends significantly on the construction of examples. These examples are necessarily intricate, and to a large extent, depend on CAZAC sequences. Finally, we present an alternative method for determining whether a Gabor system yields a tight frame. This alternative method does not prove tightness using the main theorem, but instead uses the Gram matrix of the Gabor system.

A Characterization of Projective Unitary Equivalence of Finite Frames and Applications

Many applications of finite tight frames (e.g., the use of SICs and mutually unbiased bases (MUBs) in quantum information theory and harmonic frames for the analysis of signals subject to erasures) depend only on the vectors up to projective unitary equivalence. It is well known that two finite sequences of vectors in inner product spaces are unitarily equivalent if and only if their respective inner products (Gramian matrices) are equal. Here we present a corresponding result for the projective unitary equivalence of two sequences of vectors (lines) in inner product spaces, i.e., that a finite number of (Bargmann) projective (unitary) invariants are equal. This result is equivalent to finding a rank-one matrix completion of a certain matrix. We give an algorithm to recover the sequence of vectors (up to projective unitary equivalence) from a small subset of these projective invariants and apply it to SICs, MUBs, and harmonic frames. We also extend our results to the projective similarity of vectors.

The Construction of G-Invariant Finite Tight Frames

We give a complete classification of the finite tight frames which are G-invariant, i.e., invariant under the unitary action of a group G. This result is constructive, and we use it to consider a number of examples. In particular, we determine the minimum number of generators for a tight frame for the orthogonal polynomials on an n-gon or cube, which is invariant under the symmetries of the weight.

Tight Frame with Hahn and Krawtchouk Polynomials of Several Variables

Finite tight frames for polynomial subspaces are constructed using monic Hahn polynomials and Krawtchouk polynomials of several variables. Based on these polynomial frames, two methods for constructing tight frames for the Euclidean spaces are designed. With ${\mathsf r}(d,n):= \binom{n+d-1}{n}$, the first method generates, for each $m \ge n$, two families of tight frames in ${\mathbb R}^{{\mathsf r}(d,n)}$ with ${\mathsf r}(d+1,m)$ elements. The second method generates a tight frame in ${\mathbb R}^{{\mathsf r}(d,N)}$ with $1 + N \times{\mathsf r}(d+1, N)$ vectors. All frame elements are given in explicit formulas.

Prime tight frames

We introduce a class of finite tight frames called prime tight frames and prove some of their elementary properties. In particular, we show that any finite tight frame can be written as a union of prime tight frames. We then characterize all prime harmonic tight frames and use thischaracterization to suggest effective analysis and synthesis computation strategies for such frames. Finally, we describe all prime frames constructed from the spectral tetris method, and, as a byproduct, we obtain a characterization of when the spectral tetris construction works for redundancies below two.

Constructing Finite Frames via Platonic Solids

Finite tight frames have many applications and some inter- esting physical interpretations. One of the important subjects in this area is the ways for constructing such frames. In this article we give a concrete method for constructing finite normalized frames using Platonic solids.

Random Tight Frames

We introduce probabilistic frames to study finite frames whose elements are chosen at random. While finite tight frames generalize orthonormal bases by allowing redundancy, independent, uniformly distributed points on the sphere approximately form a finite unit norm tight frame (FUNTF). In the present paper, we develop probabilistic versions of tight frames and FUNTFs to significantly weaken the requirements on the random choice of points to obtain an approximate finite tight frame. Namely, points can be chosen from any probabilistic tight frame, they do not have to be identically distributed, nor have unit norm. We also observe that classes of random matrices used in compressed sensing are induced by probabilistic tight frames.

Finite-dimensional Hilbert space and frame quantization

The quantum observables used in the case of quantum systems with finite-dimensional Hilbert space are defined either algebraically in terms of an orthonormal basis and discrete Fourier transformation or by using a continuous system of coherent states. We present an alternative approach to these important quantum systems based on the finite frame quantization. Finite systems of coherent states, usually called finite tight frames, can be defined in a natural way in the case of finite quantum systems. Novel examples of such tight frames are presented. The quantum observables used in our approach are obtained by starting from certain classical observables described by functions defined on the discrete phase space corresponding to the system. They are obtained by using a finite frame and a Klauder–Berezin–Toeplitz-type quantization. Semi-classical aspects of tight frames are studied through lower symbols of basic classical observables.

Fourier Transforms of Finite Chirps

Chirps arise in many signal processing applications. While chirps have been extensively studied as functions over both the real line and the integers, less attention has been paid to the study of chirps over finite groups. We study the existence and properties of chirps over finite cyclic groups of integers. In particular, we introduce a new definition of a finite chirp which is slightly more general than those that have been previously used. We explicitly compute the discrete Fourier transforms of these chirps, yielding results that are number-theoretic in nature. As a consequence of these results, we determine the degree to which the elements of certain finite tight frames are well distributed.

Generation of finite tight frames by Householder transformations

Finite tight frames are widely used for many applications. An important problem is to construct finite frames with prescribed norm for each vector in the tight frame. In this paper we provide a fast and simple algorithm for such a purpose. Our algorithm employs the Householder transformations. For a finite tight frame consisting of m vectors in ℝn or ℂn only O(nm) operations are needed. In addition, we also study the following question: Given a set of vectors in ℝn or ℂn, how many additional vectors, possibly with constraints, does one need to add in order to obtain a tight frame?