Diagonal Unitary Entangling Gates and Contradiagonal Quantum States
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A Finiteness Result for Circulant Core Complex Hadamard Matrices
A FINITENESS RESULT FOR CIRCULANT CORE COMPLEX HADAMARD MATRICES REMUS NICOARA AND CHASE WORLEY UNIVERSITY OF TENNESSEE, KNOXVILLE Abstract. We show that for every prime number p there exist finitely many circulant core complex Hadamard matrices of size p ` 1. The proof uses a 'derivative at infinity' argument to reduce the problem to Tao's uncertainty principle for cyclic groups of prime order (see [Tao]). 1. Introduction A complex Hadamard matrix is a matrix H P MnpCq having all entries of absolute value 1 and all rows ?1 mutually orthogonal. Equivalently, n H is a unitary matrix with all entries of the same absolute value. For ij 2πi{n example, the Fourier matrix Fn “ p! q1¤i;j¤n, ! “ e , is a Hadamard matrix. In the recent years, complex Hadamard matrices have found applications in various topics of mathematics and physics, such as quantum information theory, error correcting codes, cyclic n-roots, spectral sets and Fuglede's conjecture. A general classification of real or complex Hadamard matrices is not available. A catalogue of most known complex Hadamard matrices can be found in [TaZy]. The complete classification is known for n ¤ 5 ([Ha1]) and for self-adjoint matrices of order 6 ([BeN]). Hadamard matrices arise in operator algebras as construction data for hyperfinite subfactors. A unitary ?1 matrix U is of the form n H, H Hadamard matrix, if and only if the algebra of n ˆ n diagonal matrices Dn ˚ is orthogonal onto UDnU , with respect to the inner product given by the trace on MnpCq. Equivalently, the square of inclusions: Dn Ă MnpCq CpHq “ ¨ YY ; τ˛ ˚ ˚ ‹ ˚ C Ă UDnU ‹ ˚ ‹ ˝ ‚ is a commuting square, in the sense of [Po1],[Po2], [JS]. -
Hadamard Equiangular Tight Frames
Air Force Institute of Technology AFIT Scholar Faculty Publications 8-8-2019 Hadamard Equiangular Tight Frames Matthew C. Fickus Air Force Institute of Technology John Jasper University of Cincinnati Dustin G. Mixon Air Force Institute of Technology Jesse D. Peterson Air Force Institute of Technology Follow this and additional works at: https://scholar.afit.edu/facpub Part of the Discrete Mathematics and Combinatorics Commons Recommended Citation Fickus, M., Jasper, J., Mixon, D. G., & Peterson, J. D. (2021). Hadamard Equiangular Tight Frames. Applied and Computational Harmonic Analysis, 50, 281–302. https://doi.org/10.1016/j.acha.2019.08.003 This Article is brought to you for free and open access by AFIT Scholar. It has been accepted for inclusion in Faculty Publications by an authorized administrator of AFIT Scholar. For more information, please contact [email protected]. Hadamard Equiangular Tight Frames Matthew Fickusa, John Jasperb, Dustin G. Mixona, Jesse D. Petersona aDepartment of Mathematics and Statistics, Air Force Institute of Technology, Wright-Patterson AFB, OH 45433 bDepartment of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221 Abstract An equiangular tight frame (ETF) is a type of optimal packing of lines in Euclidean space. They are often represented as the columns of a short, fat matrix. In certain applications we want this matrix to be flat, that is, have the property that all of its entries have modulus one. In particular, real flat ETFs are equivalent to self-complementary binary codes that achieve the Grey-Rankin bound. Some flat ETFs are (complex) Hadamard ETFs, meaning they arise by extracting rows from a (complex) Hadamard matrix. -
The Early View of “Global Journal of Computer Science And
The Early View of “Global Journal of Computer Science and Technology” In case of any minor updation/modification/correction, kindly inform within 3 working days after you have received this. Kindly note, the Research papers may be removed, added, or altered according to the final status. Global Journal of Computer Science and Technology View Early Global Journal of Computer Science and Technology Volume 10 Issue 12 (Ver. 1.0) View Early Global Academy of Research and Development © Global Journal of Computer Global Journals Inc. Science and Technology. (A Delaware USA Incorporation with “Good Standing”; Reg. Number: 0423089) Sponsors: Global Association of Research 2010. Open Scientific Standards All rights reserved. Publisher’s Headquarters office This is a special issue published in version 1.0 of “Global Journal of Medical Research.” By Global Journals Inc. Global Journals Inc., Headquarters Corporate Office, All articles are open access articles distributed Cambridge Office Center, II Canal Park, Floor No. under “Global Journal of Medical Research” 5th, Cambridge (Massachusetts), Pin: MA 02141 Reading License, which permits restricted use. Entire contents are copyright by of “Global United States Journal of Medical Research” unless USA Toll Free: +001-888-839-7392 otherwise noted on specific articles. USA Toll Free Fax: +001-888-839-7392 No part of this publication may be reproduced Offset Typesetting or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information Global Journals Inc., City Center Office, 25200 storage and retrieval system, without written permission. Carlos Bee Blvd. #495, Hayward Pin: CA 94542 The opinions and statements made in this United States book are those of the authors concerned. -
Asymptotic Existence of Hadamard Matrices
ASYMPTOTIC EXISTENCE OF HADAMARD MATRICES by Ivan Livinskyi A Thesis submitted to the Faculty of Graduate Studies of The University of Manitoba in partial fulfilment of the requirements of the degree of MASTER OF SCIENCE Department of Mathematics University of Manitoba Winnipeg Copyright ⃝c 2012 by Ivan Livinskyi Abstract We make use of a structure known as signed groups, and known sequences with zero autocorrelation to derive new results on the asymptotic existence of Hadamard matrices. For any positive odd integer p it is obtained that a Hadamard matrix of order 2tp exists for all ( ) 1 p − 1 t ≥ log + 13: 5 2 2 i Contents 1 Introduction. Hadamard matrices 2 2 Signed groups and remreps 10 3 Construction of signed group Hadamard matrices from sequences 18 4 Some classes of sequences with zero autocorrelation 28 4.1 Golay sequences .............................. 30 4.2 Base sequences .............................. 31 4.3 Complex Golay sequences ........................ 33 4.4 Normal sequences ............................. 35 4.5 Turyn sequences .............................. 36 4.6 Other sequences .............................. 37 5 Asymptotic Existence results 40 5.1 Asymptotic formulas based on Golay sequences . 40 5.2 Hadamard matrices from complex Golay sequences . 43 5.3 Hadamard matrices from all complementary sequences considered . 44 6 Thoughts about further development 48 Bibliography 51 1 Chapter 1 Introduction. Hadamard matrices A Hadamard matrix H is a square (±1)-matrix such that any two of its rows are orthogonal. In other words, a square (±1)-matrix H is Hadamard if and only if HH> = nI; where n is the order of H and I is the identity matrix of order n. -
One Method for Construction of Inverse Orthogonal Matrices
ONE METHOD FOR CONSTRUCTION OF INVERSE ORTHOGONAL MATRICES, P. DIÞÃ National Institute of Physics and Nuclear Engineering, P.O. Box MG6, Bucharest, Romania E-mail: [email protected] Received September 16, 2008 An analytical procedure to obtain parametrisation of complex inverse orthogonal matrices starting from complex inverse orthogonal conference matrices is provided. These matrices, which depend on nonzero complex parameters, generalize the complex Hadamard matrices and have applications in spin models and digital signal processing. When the free complex parameters take values on the unit circle they transform into complex Hadamard matrices. 1. INTRODUCTION In a seminal paper [1] Sylvester defined a general class of orthogonal matrices named inverse orthogonal matrices and provided the first examples of what are nowadays called (complex) Hadamard matrices. A particular class of inverse orthogonal matrices Aa()ij are those matrices whose inverse is given 1 t by Aa(1ij ) (1 a ji ) , where t means transpose, and their entries aij satisfy the relation 1 AA nIn (1) where In is the n-dimensional unit matrix. When the entries aij take values on the unit circle A–1 coincides with the Hermitian conjugate A* of A, and in this case (1) is the definition of complex Hadamard matrices. Complex Hadamard matrices have applications in quantum information theory, several branches of combinatorics, digital signal processing, etc. Complex orthogonal matrices unexpectedly appeared in the description of topological invariants of knots and links, see e.g. Ref. [2]. They were called two- weight spin models being related to symmetric statistical Potts models. These matrices have been generalized to two-weight spin models, also called , Paper presented at the National Conference of Physics, September 10–13, 2008, Bucharest-Mãgurele, Romania. -
Introduction to Linear Algebra
Introduction to linear algebra Teo Banica Real matrices, The determinant, Complex matrices, Calculus and applications, Infinite matrices, Special matrices 08/20 Foreword These are slides written in the Fall 2020, on linear algebra. Presentations available at my Youtube channel. 1. Real matrices and their properties ... 3 2. The determinant of real matrices ... 19 3. Complex matrices and diagonalization ... 35 4. Linear algebra and calculus questions ... 51 5. Infinite matrices and spectral theory ... 67 6. Special matrices and matrix tricks ... 83 Real matrices and their properties Teo Banica "Introduction to linear algebra", 1/6 08/20 Rotations 1/3 Problem: what’s the formula of the rotation of angle t? Rotations 2/3 2 x The points in the plane R can be represented as vectors y . The 2 × 2 matrices “act” on such vectors, as follows: a b x ax + by = c d y cx + dy Many simple transformations (symmetries, projections..) can be written in this form. What about the rotation of angle t? Rotations 3/3 A quick picture shows that we must have: ∗ ∗ 1 cos t = ∗ ∗ 0 sin t Also, by paying attention to positives and negatives: ∗ ∗ 0 − sin t = ∗ ∗ 1 cos t Thus, the matrix of our rotation can only be: cos t − sin t R = t sin t cos t By "linear algebra”, this is the correct answer. Linear maps 1/4 2 2 Theorem. The maps f : R ! R which are linear, in the sense that they map lines through 0 to lines through 0, are: x ax + by f = y cx + dy Remark. If we make the multiplication convention a b x ax + by = c d y cx + dy the theorem says f (v) = Av, with A being a 2 × 2 matrix. -
The 2-Transitive Complex Hadamard Matrices
undergoing revision for Designs, Codes and Cryptography 1 January, 2001 The 2-Transitive Complex Hadamard Matrices G. Eric Moorhouse Dept. of Mathematics, University of Wyoming, Laramie WY, U.S.A. Abstract. We determine all possibilities for a complex Hadamard ma- trix H admitting an automorphism group which permutes 2-transitively the rows of H. Our proof of this result relies on the classification theo- rem for finite 2-transitive permutation groups, and thereby also on the classification of finite simple groups. Keywords. complex Hadamard matrix, 2-transitive, distance regular graph, cover of complete bipartite graph 1. Introduction Let H be a complex Hadamard matrix of order v, i.e. a v × v matrix whose entries are com- plex roots of unity, satisfying HH∗ = vI,where∗ denotes conjugate-transpose. Butson’s original definition in [3] considered as entries only p-th roots of unity for some prime p, while others [42], [32], [8] have considered ±1, ±i as entries. These are generalisations of the (ordinary) Hadamard matrices (having entries ±1); other generalisations with group entries are described in Section 2. 1.1 Example. The complex Hadamard matrix ⎡ ⎤ 111111 ⎢ 2 2 ⎥ ⎢ 11 ωω ω ω ⎥ ⎢ 2 2 ⎥ ⎢ 1 ω 1 ωω ω ⎥ 2πi/3 H6 = ⎢ 2 2 ⎥ ,ω= e ⎣ 1 ω ω 1 ωω⎦ 1 ω2 ω2 ω 1 ω 1 ωω2 ω2 ω 1 of order 6 corresponds (as in Section 2) to the distance transitive triple cover of the complete bipartite graph K6,6 which appears in [2, Thm. 13.2.2]. ∗ An automorphism of H is a pair (M1 ,M2) of monomial matrices such that M1HM2 = H. -
LOGIC JOURNAL of the IGPL
LOGIC JOURNAL LOGIC ISSN PRINT 1367-0751 LOGIC JOURNAL ISSN ONLINE 1368-9894 of the IGPL LOGIC JOURNAL Volume 22 • Issue 1 • February 2014 of the of the Contents IGPL Original Articles A Routley–Meyer semantics for truth-preserving and well-determined Łukasiewicz 3-valued logics 1 IGPL Gemma Robles and José M. Méndez Tarski’s theorem and liar-like paradoxes 24 Ming Hsiung Verifying the bridge between simplicial topology and algebra: the Eilenberg–Zilber algorithm 39 L. Lambán, J. Rubio, F. J. Martín-Mateos and J. L. Ruiz-Reina INTEREST GROUP IN PURE AND APPLIED LOGICS Reasoning about constitutive norms in BDI agents 66 N. Criado, E. Argente, P. Noriega and V. Botti 22 Volume An introduction to partition logic 94 EDITORS-IN-CHIEF David Ellerman On the interrelation between systems of spheres and epistemic A. Amir entrenchment relations 126 • D. M. Gabbay Maurício D. L. Reis 1 Issue G. Gottlob On an inferential semantics for classical logic 147 David Makinson R. de Queiroz • First-order hybrid logic: introduction and survey 155 2014 February J. Siekmann Torben Braüner Applications of ultraproducts: from compactness to fuzzy elementary classes 166 EXECUTIVE EDITORS Pilar Dellunde M. J. Gabbay O. Rodrigues J. Spurr www.oup.co.uk/igpl JIGPAL-22(1)Cover.indd 1 16-01-2014 19:02:05 Logical Information Theory: New Logical Foundations for Information Theory [Forthcoming in: Logic Journal of the IGPL] David Ellerman Philosophy Department, University of California at Riverside June 7, 2017 Abstract There is a new theory of information based on logic. The definition of Shannon entropy as well as the notions on joint, conditional, and mutual entropy as defined by Shannon can all be derived by a uniform transformation from the corresponding formulas of logical information theory. -
What Can Logic Contribute to Information Theory?
What can logic contribute to information theory? David Ellerman University of California at Riverside September 10, 2016 Abstract Logical probability theory was developed as a quantitative measure based on Boole’s logic of subsets. But information theory was developed into a mature theory by Claude Shannon with no such connection to logic. But a recent development in logic changes this situation. In category theory, the notion of a subset is dual to the notion of a quotient set or partition, and recently the logic of partitions has been developed in a parallel relationship to the Boolean logic of subsets (subset logic is usually mis-specified as the special case of propositional logic). What then is the quantitative measure based on partition logic in the same sense that logical probability theory is based on subset logic? It is a measure of information that is named "logical entropy" in view of that logical basis. This paper develops the notion of logical entropy and the basic notions of the resulting logical information theory. Then an extensive comparison is made with the corresponding notions based on Shannon entropy. Contents 1 Introduction 2 2 Duality of subsets and partitions 3 3 Classical subset logic and partition logic 4 4 Classical probability and classical logical entropy 5 5 History of logical entropy formula 7 6 Comparison as a measure of information 8 7 The dit-bit transform 9 8 Conditional entropies 10 8.1 Logical conditional entropy . 10 8.2 Shannon conditional entropy . 13 9 Mutual information 15 9.1 Logical mutual information for partitions . 15 9.2 Logical mutual information for joint distributions . -
On Quaternary Complex Hadamard Matrices of Small Orders
Advances in Mathematics of Communications doi:10.3934/amc.2011.5.309 Volume 5, No. 2, 2011, 309{315 ON QUATERNARY COMPLEX HADAMARD MATRICES OF SMALL ORDERS Ferenc Szoll¨ osi} Department of Mathematics and its Applications Central European University H-1051, N´adoru. 9, Budapest, Hungary (Communicated by Gilles Z´emor) Abstract. One of the main goals of design theory is to classify, character- ize and count various combinatorial objects with some prescribed properties. In most cases, however, one quickly encounters a combinatorial explosion and even if the complete enumeration of the objects is possible, there is no ap- parent way how to study them in details, store them efficiently, or generate a particular one rapidly. In this paper we propose a novel method to deal with these difficulties, and illustrate it by presenting the classification of quaternary complex Hadamard matrices up to order 8. The obtained matrices are mem- bers of only a handful of parametric families, and each inequivalent matrix, up to transposition, can be identified through its fingerprint. 1. Introduction A complex Hadamard matrix H of order n is an n × n matrix with unimodular entries satisfying HH∗ = nI where ∗ is the conjugate transpose and I is the iden- tity matrix of order n. In other words, any two distinct rows (or columns) of H are complex orthogonal. Complex Hadamard matrices have important applications in quantum optics, high-energy physics [1], operator theory [16] and in harmonic analysis [12], [20]. They also play a crucial r^olein quantum information theory, for construction of teleportation and dense coding schemes [21], and they are strongly related to mutually unbiased bases (MUBs) [8]. -
Almost Hadamard Matrices with Complex Entries
Adv. Oper. Theory 3 (2018), no. 1, 137–177 http://doi.org/10.22034/aot.1702-1114 ISSN: 2538-225X (electronic) http://aot-math.org ALMOST HADAMARD MATRICES WITH COMPLEX ENTRIES TEODOR BANICA,1∗ and ION NECHITA2 Dedicated to the memory of Uffe Haagerup Communicated by N. Spronk Abstract. We discuss an extension of the almost Hadamard matrix formal- ism, to the case of complex matrices. Quite surprisingly, the situation here is very different from the one in the real case, and our conjectural conclusion is that there should be no such matrices, besides the usual Hadamard ones. We verify this conjecture in a number of situations, and notably for most of the known examples of real almost Hadamard matrices, and for some of their complex extensions. We discuss as well some potential applications of our conjecture, to the general study of complex Hadamard matrices. Introduction An Hadamard matrix is a square matrix H ∈ MN (±1), whose rows are pairwise orthogonal. Here is a basic example, which appears as a version of the 4×4 Walsh matrix, and is also well-known for its use in the Grover algorithm: −1 1 1 1 1 −1 1 1 K4 = 1 1 −1 1 1 1 1 −1 Assuming that the matrix has N ≥ 3 rows, the orthogonality conditions be- tween the rows give N ∈ 4N. A similar analysis with four or more rows, or any Copyright 2016 by the Tusi Mathematical Research Group. Date: Received: Feb. 9, 2017; Accepted: May 12, 2017. ∗Corresponding author. 2010 Mathematics Subject Classification. Primary 15B10; Secondary 05B20, 14P05. -
The Quaternary Complex Hadamard Matrices of Orders 10, 12, and 14
Discrete Mathematics 313 (2013) 189–206 Contents lists available at SciVerse ScienceDirect Discrete Mathematics journal homepage: www.elsevier.com/locate/disc The quaternary complex Hadamard matrices of orders 10, 12, and 14 Pekka H.J. Lampio a,∗, Ferenc Szöllősi b, Patric R.J. Östergård a a Department of Communications and Networking, Aalto University School of Electrical Engineering, P.O. Box 13000, 00076 Aalto, Finland b Department of Mathematics and its Applications, Central European University, H-1051, Nádor u. 9, Budapest, Hungary article info a b s t r a c t Article history: A complete classification of quaternary complex Hadamard matrices of orders 10, 12 and Received 4 March 2012 14 is given, and a new parametrization scheme for obtaining new examples of affine Received in revised form 29 September parametric families of complex Hadamard matrices is provided. On the one hand, it is 2012 proven that all 10 × 10 and 12 × 12 quaternary complex Hadamard matrices belong to Accepted 1 October 2012 some parametric family, but on the other hand, it is shown by exhibiting an isolated 14×14 Available online 31 October 2012 matrix that there cannot be a general method for introducing parameters into these types of matrices. Keywords: Butson-type Hadamard matrix ' 2012 Elsevier B.V. All rights reserved. Classification Complex Hadamard matrix 1. Introduction A complex Hadamard matrix H of order n is an n × n matrix with unimodular entries satisfying HH∗ D nI where ∗ is the conjugate transpose, I is the identity matrix of order n, and where unimodular means that the entries are complex numbers that lie on the unit circle.