Frames over finite fields: Basic theory and equiangular lines in unitary geometry

Finite tight frames play an important role in miscellaneous areas, including quantum information theory. Here we apply a class of tight frames, equiangular tight frames, to address the problem of detecting the entanglement of bipartite states. Here we derive some entanglement criteria based on positive operator-valued measurements built from equiangular tight frames. We also present a class of entanglement witnesses based on the equiangular tight frames. At last, we generalize the entanglement criterion for bipartite systems to multipartite systems.

Equiangular tight frames and fourth root seidel matrices

We present the theory of equiangular tight frames (ETFs) from association schemes as a generalization of harmonic ETFs. This theory has the pleasing consequence of including not only harmonic ETFs from abelian groups but also real ETFs with centroidal symmetry and the ETFs discovered by Renes and Strohmer coming from skew-symmetric conference matrices. Finally we present a theory of nonabelian harmonic ETFs from the perspective of association schemes. An infinite class of ETFs arising as group frames from nonabelian groups is presented.

An equiangular tight frame (ETF) is an M N matrix which has orthogonal equal norm rows, equal normcolumns, and the inner products of all pairs of columns have the same modulus. In this paper we studyETFs in which all of the entries are unimodular, and in particular pth roots of unity. A new constructionof unimodular ETFs based on resolvable Steiner systems is presented. This construction gives many newexamples of unimodular ETFs. In particular, an new in nite class of ETFs with entries in f1; 1g ispresented.Keywords: tight frames, equiangular, Hadamard matrices, Steiner systems, resolvable 1. INTRODUCTION Let F be an M N matrix with real or complex entries. If there exists A > 0 such that FF = AI , where Iis the identity matrix, then F is called a tight frame . If F F has constant diagonal and all of the o -diagonalentries have the same modulus, then F is called equiangular. If all of the entries of F have modulus 1 thenwe call F unimodular . If all of the entries of F are pth roots of unity, then we call F unimodular of degree p.This article is concerned with equiangular tight frames (ETFs) which are unimodular of degree p. Inparticular, we are concerned with nding unimodular ETFs of degree 2, which we shall also refer to asHadamard ETFs. Note that in the usual de nition of ETFs the columns are taken to have norm 1. However,since we are concerned with unimodular ETFs we will avoid normalizing, thus the columns of an M Nunimodular ETF have norm M .The paper is organized as follows. In Section 2 we present some necessary conditions on the existence ofunimodular ETFs. In Section 3 we review the construction of harmonic ETFs, which is an important classof ETFs that also happen to be unimodular. In Section 4 we review the construction of the Steiner ETFs ofFickus, Mixon, and Tremain,

Detecting or classifying already known sparse signals contaminated by Gaussian noise from compressive measurements is different from reconstructing sparse signals, as its objective is to minimize the error probability which describes performance of the detectors or classifiers. This paper is concerned about the performance improvement of a commonly used Compressive Classifier. We prove that when the arbitrary sensing matrices used to get the Compressive Measurements are transformed into Equi-Norm Tight Frames, i.e. the matrices that are row-orthogonal, The Compressive Classifier achieves better performance. Although there are other proofs that among all Equi-Norm Tight Frames the Equiangular tight Frames (ETFs) bring best worst-case performance, the existence and construction of ETFs on some dimensions is still an open problem. As the construction of Equi-Norm Tight Frames from any arbitrary matrices is very easy and practical compared with ETF matrices, the result of this paper can also provide a practical method to design an improved sensing matrix for Compressive Classification. We can conclude that: Tight is Better!

Let A be a subset of a finite field $$ F := \mathbf{Z}/q\mathbf{Z} $$ for some prime q. If $$ |F|^{\delta} < |A| < |F|^{1-\delta} $$ for some δ > 0, then we prove the estimate $$ |A + A| + |A \cdot A| \geq c(\delta)|A|^{1+\varepsilon} $$ for some ε = ε(δ) > 0. This is a finite field analogue of a result of [ErS]. We then use this estimate to prove a Szemerédi-Trotter type theorem in finite fields, and obtain a new estimate for the Erdös distance problem in finite fields, as well as the three-dimensional Kakeya problem in finite fields.

The relationship between equiangular tight frames and strongly regular graphs has been known for several years. This relationship has been exploited to construct many of the latest examples of new strongly regular graphs. Recently it was shown that there is a similar relationship between two-distance tight frames and strongly regular graphs. In this paper we present a new tensor like construction of two-distance tight frames, and hence a family of strongly regular graphs. While graphs with these parameters were known to exist, this new construction is very simple, requiring only the existence of an affine plane, whereas the original constructions often require more complicated objects such as generalized quadrangles.

A complex equiangular tight frame (ETF) is a tight frame consisting of N unit vectors in C d whose absolute inner products are identical. One may view complex ETFs as a natural geometric generalization of an orthonormal basis. Numerical evidence suggests that these objects do not arise for most pairs ( d , N). The goal of this paper is to develop conditions on ( d , N) under which complex ETFs can exist. In particular, this work concentrates on the class of harmonic ETFs, in which the components of the frame vectors are roots of unity. In this case, it is possible to leverage field theory to obtain stringent restrictions on the possible values for ( d , N).

Inspired by the works of L. Carlitz [Acta Arith. 5 (1959), pp. 293–308] and Z.-W. Sun [Finite Fields Appl. 56 (2019), pp. 285–307] on cyclotomic matrices, in this paper, we investigate certain cyclotomic matrices involving Gauss sums over finite fields, which can be viewed as finite field analogues of certain matrices related to the Gamma function. For example, let q = p n q=p^n be an odd prime power with p p prime and n ∈ Z + n\in \mathbb {Z}^+ . Let ζ p = e 2 π i / p \zeta _p=e^{2\pi \mathbf {i}/p} and let χ \chi be a generator of the group of all multiplicative characters of the finite field F q \mathbb {F}_q . For the Gauss sum G q ( χ r ) = ∑ x ∈ F q χ r ( x ) ζ p T r F q / F p ( x ) , \begin{equation*} G_q(\chi ^{r})=\sum _{x\in \mathbb {F}_q}\chi ^{r}(x)\zeta _p^{\mathrm {Tr}_{\mathbb {F}_q/\mathbb {F}_p}(x)}, \end{equation*} we prove that det [ G q ( χ 2 i + 2 j ) ] 0 ≤ i , j ≤ ( q − 3 ) / 2 = ( − 1 ) α n ( q − 1 2 ) q − 1 2 2 p n − 1 − 1 2 , \begin{equation*} \det \left [G_q(\chi ^{2i+2j})\right ]_{0\le i,j\le (q-3)/2}=(-1)^{\alpha _n}\left (\frac {q-1}{2}\right )^{\frac {q-1}{2}}2^{\frac {p^{n-1}-1}{2}}, \end{equation*} where α n = { 1 a m p ; if n ≡ 1 ( mod 2 ) , ( p 2 + 7 ) / 8 a m p ; if n ≡ 0 ( mod 2 ) . \begin{equation*} \alpha _n= \begin {cases} 1 &amp; \text {if } n\equiv 1\pmod 2,\\ (p^2+7)/8 &amp; \text {if } n\equiv 0\pmod 2. \end{cases} \end{equation*}

From the 19th century, the theory of permutation polynomial over finite fields, that are arose in the work of Hermite and Dickson, has drawn general attention. Permutation polynomials over finite fields are an active area of research due to their rising applications in mathematics and engineering. The last three decades has seen rapid progress on the research on permutation polynomials due to their diverse applications in cryptography, coding theory, finite geometry, combinatorics and many more areas of mathematics and engineering. For this reason, the study of permutation polynomials is important nowadays. In this chapter, we propose some new problems in connection to permutation polynomials over finite fields by the help of prime numbers.

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.

Review for "Entropic uncertainty relations from equiangular tight frames and their applications"

Decision letter for "Entropic uncertainty relations from equiangular tight frames and their applications"

One of the most celebrated applications of Gauss’s 2 F 1 _2F_1 hypergeometric functions is in connection with the rapid convergence of sequences and special values that arise in the theory of arithmetic and geometric means. This theory was the inspiration for a recent paper (see M. J. Griffin, Ken Ono, N. Saikia, and W.-L. Tsai, [AGM and jellyfish swarms of elliptic curves, Amer. Math. Monthly 130 (2023), no. 4, 355-369]) in which a finite field analogue of A G M R \mathrm {AGM}_\mathbb {R} was defined and then studied using finite field hypergeometric functions. Instead of convergent sequences, one gets directed graphs that combine to form disjoint unions of graphs that individually resemble jellyfish. Echoing the connection of hypergeometric functions to periods of elliptic curves, these graphs organize elliptic curves over finite fields. Here we use such “jellyfish swarms” to prove new identities for Gauss’s class numbers of positive definite binary quadratic forms. Moreover, we prove that the sizes of jellyfish are in part dictated by the order of the prime above 2 in certain class groups.

Finite fields play important roles in many application areas such as coding theory, cryptography, Monte Carlo and quasi-Monte Carlo methods, pseudorandom number generation, quantum computing, and wireless communication. In this book we will focus on sequences, character sums, and polynomials over finite fields in view of the above mentioned application areas: Chapters 1 and 2 deal with sequences mainly constructed via characters and analyzed using bounds on character sums. Chapters 3, 5, and 6 deal with polynomials over finite fields. Chapters 4 and 9 consider problems related to coding theory studied via finite geometry and additive combinatorics, respectively. Chapter 7 deals with quasirandom points in view of applications to numerical integration using quasi-Monte Carlo methods and simulation. Chapter 8 studies aspects of iterations of rational functions from which pseudorandom numbers for Monte Carlo methods can be derived. The goal of this book is giving an overview of several recent research directions as well as stimulating research in sequences and polynomials under the unified framework of character theory.