DOCSLIB.ORG

Conference Matrices and Unimodular Lattices

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Conference Matrices and Unimodular Lattices View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Europ. J. Combinatorics (2001) 22, 1033–1045 doi:10.1006/eujc.2001.0539 Available online at http://www.idealibrary.com on Conference Matrices and Unimodular Lattices ROBIN CHAPMAN Conference matrices are used to define complex structures on real vector spaces. Certain lattices in these spaces become modules for rings of quadratic integers. Multiplication of these lattices by nonprincipal ideals yields simple constructions of further lattices including the Leech lattice. c 2001 Academic Press 1. INTRODUCTION We use conference matrices to define an action of the complex numbers on the real Euclidean n vector space R . In certain cases, the lattice DnC becomes a module over a ring of quadratic in- tegers. We can then obtain new unimodular lattices, essentially by multiplying the lattice DnC by a nonprincipal ideal in this ring. We show that lattices constructed via quadratic residue codes, including the Leech lattice, can be constructed in this way. Recall that a lattice 3 is a discrete subgroup of a finite dimensional real vector space V . We suppose that V has a given Euclidean inner product .u; v/ u v and the rank of 3 equals the dimension of V . In this case 3 has a bounded fundamental7! · region in V . We call the volume of such a fundamental region (measured with respect to the Euclidean structure on V ) the volume of the lattice 3. The lattice 3 is integral if u v Z for all u, v 3. It is even if u 2 u u 2Z for all u 3. Even lattices are necessarily· 2 integral. The lattice2 3 is unimodularj j Dif 3· is2 integral and has2 volume 1. It is well known [9, Chapter VIII, Theorem 8] that if 3 is an even unimodular then the rank of 3 is divisible by 8. For convenience we call the square of the length of a vector its norm. The minimum norm of a lattice is the smallest nonzero norm of its vectors. 2. CONFERENCE MATRICES Let l be a positive integer. A conference matrix of order n [7, Chapter 18] is an n-by-n matrix W satisfying (a) the diagonal entries of W vanish, while its off-diagonal entries lie in 1; 1 , (b) WW .n 1/I , where I denotes the n-by-n identity matrix. {− g > D − Let n denote the set of skew-symmetric conference matrices of order n. W t 2 Let W n. Then H I W satisfies HH .I W/.I W/ I W 2 W D C D C − D − D I WW > nI . As all the entries of H lie in 1; 1 then H is a Hadamard matrix. ConsequentlyC D [7, Theorem 18.1] n 1, 2 or is a multiple{− ofg 4. D n Suppose that n is a multiple of 4 and let l n 1. Fix W n and let V R denote the n-dimensional real vector space under the standardD − Euclidean2 W dot product. WeD shall make V into a vector space over the complex numbers. Each complex number z has the form r sp l where r and s are real numbers and p l ipl. We then define C − − D r sp l v v.r I sW/ rv svW C − D C D C for each v V . Then, since W 2 l I , it is straightforward to show that V is now a complex vector space,2 that is D − 0195–6698/01/081033 + 13 $35.00/0 c 2001 Academic Press 1034 R. Chapman (a) z.v w/ zv zw for z C, v, w V ; C D C 2 2 (b) .z1 z2/v z1v z2v and .z1z2/v z1.z2v/ for z1, z2 C, v V . C D C D 2 2 As a complex vector space, V has dimension n=2. Let v pv v denote the Euclidean length of a vector v Rn. This action of C on n j j D · 2 R transforms lengths in a simple way. Let z∗ denote the complex conjugate of the complex number z. n LEMMA 2.1. (a) If z1, z2 C and v1, v2 R then .z1v1/ .z2v2/ .z1z2∗v1/ v2 (b) If z C and v Rn then2zv z v . 2 · D · 2 2 j j D j jj j PROOF. Let z j r j s j p l with r j , s j R. Then D C − 2 .z1v1/ .z2v2/ .z1v1/.z2v2/> · D v1.r1 I s1W/.r2 I s2W/>v> D C C 2 v.r1 I s1W/.r2 I s2W/v> D C − v..r1r2 ls1s2/I .s1r2 r1s2/W/v> D C C − .z1z∗v1/ v2 D 2 · as claimed. Consequently 2 2 2 2 zv .zv/ .zv/ .zz∗v/ v z v v z v : 2 j j D · D · D j j · D j j j j Thus for fixed nonzero z, the map v zv is a similarity of Rn with scale factor z . 7! j j 3. QUADRATIC FIELDS We retain the previous notation. Suppose in addition that l n 1 is square free. Let K denote the quadratic field Q p l . Since l is square free, the ringD of− integers of K is − 1 p l a bp l Z " C − # ( C − a; b Z; a b .mod 2/) : O D 2 D 2 V 2 ≡ In particular is a Dedekind domain. We shall show that some of the familiar lattices L in l 1 O R C are modules for the ring , that is that αL L whenever α . Let O ⊆ 2 O n L0 .a1;:::; an/ Z a1 an 0 .mod 2/ D f 2 V C···C ≡ g be the Dn root lattice. LEMMA 3.1. The lattice L0 is an -module. O 1 1 PROOF. It suffices to show that 1 p l v v.I W/ L0 whenever v L0. Indeed 2 C − D 2 C 2 2 it suffices to show this whenever v lies in a generating set for L0. Now L0 is generated by the vectors 2e j and e j ek (for j k) where e j denotes the jth unit vector. Firstly e j .I W/ is a row of the HadamardC matrix6DI W. As it contains n instances of 1 and n is even,C it lies 1 C ± in L0. Next 2 .e j ek/.I W/ is the sum of two rows of the Hadamard matrix I W. Two C C 1 C rows of an n-by-n Hadamard matrix agree in exactly n=2 places. Hence 2 .e j ek/.I W/ 1 C C has n=2 zeros and n=2 instances of 1. As n=2 is even then 2 .e j ek/.I W/ L0. This completes the proof. ± C C 2 2 Unimodular lattices 1035 Now consider the set .a1;:::; an/ a j 1=2; 1=2 : S D f V 2 {− gg The difference of two vectors in lies in L0 if and only if those vectors agree in an even S number of places. Thus there are exactly two cosets v L0 as v runs through . The union of with either of these cosets is a lattice and it will beC convenient to distinguishS these lattices byS a notation dependent on the matrix W. 1 1 For each j, 2 e j .I W/ , and for each j and k, 2 .e j ek/.I W/ L0 by Lemma 3.1. 1 C 2 S − C 2 Thus the cosets e j .I W/ L0 are identical. Let 2 C C 1 v v e1.I W/ L0 SC D f 2 S V − 2 C 2 g and : S− D S n SC 1 1 1 As e j .I W/ e j . I W/ e j = L0 then e j . I W/ for each j. 2 C − 2 − C D 2 2 − C 2 S− If v then 2v has n entries 1 and so 2v L0. It follows that L0 .v L0/ is a lattice, 2 S ± 2 [ C which depends only on whether v or v . We write L for L0 .v L0/ when 2 SC 2 S− C [ C v and L for L0 .v L0/ when v . Both L and L are isometric to the lattice 2 SC − [ C 2 S− C − usually denoted by DnC [5, Chapter 4, Section 7.3]. The lattice DnC is unimodular for each n divisible by 4, and it is even unimodular whenever n is divisible by 8. LEMMA 3.2. If n is divisible by 8 then the lattices L and L are -modules. C − O PROOF. Let L L or L . Then L L0 .v L0/ for some v and by Lemma 3.1 D C1 − D1 C C 1 2 S it suffices to show that 2 1 p l v 2 v.I W/ L. Note that 4 .l 1/ is an even integer by the hypothesis. C − D C 2 C 1 1 We may assume that v e1. I W/. If v e1.I W/ then D 2 ± C D 2 C 1 1 2 1 1 l 1 v.I W/ e1.I W/ e1..1 l/I 2W/ e1.I W/ C e1 2 C D 4 C D 4 − C D 2 C − 4 1 which lies in L as 2 e1.I W/ L. 1 C 2 If v e1.
Load more

Recommended publications
  • Codes and Modules Associated with Designs and T-Uniform Hypergraphs Richard M
    Codes and modules associated with designs and t-uniform hypergraphs Richard M. Wilson California Institute of Technology (1) Smith and diagonal form (2) Solutions of linear equations in integers (3) Square incidence matrices (4) A chain of codes (5) Self-dual codes; Witt’s theorem (6) Symmetric and quasi-symmetric designs (7) The matrices of t-subsets versus k-subsets, or t-uniform hy- pergaphs (8) Null designs (trades) (9) A diagonal form for Nt (10) A zero-sum Ramsey-type problem (11) Diagonal forms for matrices arising from simple graphs 1. Smith and diagonal form Given an r by m integer matrix A, there exist unimodular matrices E and F ,ofordersr and m,sothatEAF = D where D is an r by m diagonal matrix. Here ‘diagonal’ means that the (i, j)-entry of D is 0 unless i = j; but D is not necessarily square. We call any matrix D that arises in this way a diagonal form for A. As a simple example, ⎛ ⎞ 0 −13 10 314⎜ ⎟ 100 ⎝1 −1 −1⎠ = . 21 4 −27 050 01−2 The matrix on the right is a diagonal form for the middle matrix on the left. Let the diagonal entries of D be d1,d2,d3,.... ⎛ ⎞ 20 0 ··· ⎜ ⎟ ⎜0240 ···⎟ ⎜ ⎟ ⎝0 0 120 ···⎠ . ... If all diagonal entries di are nonnegative and di divides di+1 for i =1, 2,...,thenD is called the integer Smith normal form of A, or simply the Smith form of A, and the integers di are called the invariant factors,ortheelementary divisors of A.TheSmith form is unique; the unimodular matrices E and F are not.
    [Show full text]
  • Hadamard and Conference Matrices
    Hadamard and conference matrices Peter J. Cameron December 2011 with input from Dennis Lin, Will Orrick and Gordon Royle Now det(H) is equal to the volume of the n-dimensional parallelepiped spanned by the rows of H. By assumption, each row has Euclidean length at most n1/2, so that det(H) ≤ nn/2; equality holds if and only if I every entry of H is ±1; > I the rows of H are orthogonal, that is, HH = nI. A matrix attaining the bound is a Hadamard matrix. Hadamard's theorem Let H be an n × n matrix, all of whose entries are at most 1 in modulus. How large can det(H) be? A matrix attaining the bound is a Hadamard matrix. Hadamard's theorem Let H be an n × n matrix, all of whose entries are at most 1 in modulus. How large can det(H) be? Now det(H) is equal to the volume of the n-dimensional parallelepiped spanned by the rows of H. By assumption, each row has Euclidean length at most n1/2, so that det(H) ≤ nn/2; equality holds if and only if I every entry of H is ±1; > I the rows of H are orthogonal, that is, HH = nI. Hadamard's theorem Let H be an n × n matrix, all of whose entries are at most 1 in modulus. How large can det(H) be? Now det(H) is equal to the volume of the n-dimensional parallelepiped spanned by the rows of H. By assumption, each row has Euclidean length at most n1/2, so that det(H) ≤ nn/2; equality holds if and only if I every entry of H is ±1; > I the rows of H are orthogonal, that is, HH = nI.
    [Show full text]
  • 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.
    [Show full text]
  • Hadamard Matrices Include
    Hadamard and conference matrices Peter J. Cameron University of St Andrews & Queen Mary University of London Mathematics Study Group with input from Rosemary Bailey, Katarzyna Filipiak, Joachim Kunert, Dennis Lin, Augustyn Markiewicz, Will Orrick, Gordon Royle and many happy returns . Happy Birthday, MSG!! Happy Birthday, MSG!! and many happy returns . Now det(H) is equal to the volume of the n-dimensional parallelepiped spanned by the rows of H. By assumption, each row has Euclidean length at most n1/2, so that det(H) ≤ nn/2; equality holds if and only if I every entry of H is ±1; > I the rows of H are orthogonal, that is, HH = nI. A matrix attaining the bound is a Hadamard matrix. This is a nice example of a continuous problem whose solution brings us into discrete mathematics. Hadamard's theorem Let H be an n × n matrix, all of whose entries are at most 1 in modulus. How large can det(H) be? A matrix attaining the bound is a Hadamard matrix. This is a nice example of a continuous problem whose solution brings us into discrete mathematics. Hadamard's theorem Let H be an n × n matrix, all of whose entries are at most 1 in modulus. How large can det(H) be? Now det(H) is equal to the volume of the n-dimensional parallelepiped spanned by the rows of H. By assumption, each row has Euclidean length at most n1/2, so that det(H) ≤ nn/2; equality holds if and only if I every entry of H is ±1; > I the rows of H are orthogonal, that is, HH = nI.
    [Show full text]
  • Lattices from Tight Equiangular Frames Albrecht Böttcher
    Claremont Colleges Scholarship @ Claremont CMC Faculty Publications and Research CMC Faculty Scholarship 1-1-2016 Lattices from tight equiangular frames Albrecht Böttcher Lenny Fukshansky Claremont McKenna College Stephan Ramon Garcia Pomona College Hiren Maharaj Pomona College Deanna Needell Claremont McKenna College Recommended Citation Albrecht Böttcher, Lenny Fukshansky, Stephan Ramon Garcia, Hiren Maharaj and Deanna Needell. “Lattices from equiangular tight frames.” Linear Algebra and its Applications, vol. 510, 395–420, 2016. This Article is brought to you for free and open access by the CMC Faculty Scholarship at Scholarship @ Claremont. It has been accepted for inclusion in CMC Faculty Publications and Research by an authorized administrator of Scholarship @ Claremont. For more information, please contact [email protected]. Lattices from tight equiangular frames Albrecht B¨ottcher, Lenny Fukshansky, Stephan Ramon Garcia, Hiren Maharaj, Deanna Needell Abstract. We consider the set of all linear combinations with integer coefficients of the vectors of a unit tight equiangular (k, n) frame and are interested in the question whether this set is a lattice, that is, a discrete additive subgroup of the k-dimensional Euclidean space. We show that this is not the case if the cosine of the angle of the frame is irrational. We also prove that the set is a lattice for n = k + 1 and that there are infinitely many k such that a lattice emerges for n = 2k. We dispose of all cases in dimensions k at most 9. In particular, we show that a (7,28) frame generates a strongly eutactic lattice and give an alternative proof of Roland Bacher’s recent observation that this lattice is perfect.
    [Show full text]
  • SYMMETRIC CONFERENCE MATRICES of ORDER Pq2 + 1
    Can. J. Math., Vol. XXX, No. 2, 1978, pp. 321-331 SYMMETRIC CONFERENCE MATRICES OF ORDER pq2 + 1 RUDOLF MATHON Introduction and definitions. A conference matrix of order n is a square matrix C with zeros on the diagonal and zbl elsewhere, which satisfies the orthogonality condition CCT = (n — 1)1. If in addition C is symmetric, C = CT, then its order n is congruent to 2 modulo 4 (see [5]). Symmetric conference matrices (C) are related to several important combinatorial configurations such as regular two-graphs, equiangular lines, Hadamard matrices and balanced incomplete block designs [1; 5; and 7, pp. 293-400]. We shall require several definitions. A strongly regular graph with parameters (v, k, X, /z) is an undirected regular graph of order v and degree k with adjacency matrix A = AT satisfying (1.1) A2 + 0* - \)A + (/x - k)I = \xJ, AJ = kJ where / is the all one matrix. Note that in a strongly regular graph any two adjacent (non-adjacent) vertices are adjacent to exactly X(/x) other vertices. An easy counting argument implies the following relation between the param­ eters v, k, X and /x; (1.2) k(k - X - 1) = (v - k - l)/x. A strongly regular graph is said to be pseudo-cyclic (PC) if v — 1 = 2k = 4/x. From (1.1) and (1.2) it is readily deduced that a PC-graph has parameters of the form (4/ + 1, 2/, t — 1, /), where t > 0 is an integer. We note that the complement of a PC-graph is again pseudo-cyclic with the same parameters as the original graph (though not necessarily isomorphic to it), i.e.
    [Show full text]
  • MATRIX THEORY and APPLICATIONS Edited by Charles R
    http://dx.doi.org/10.1090/psapm/040 AMS SHORT COURSE LECTURE NOTES Introductory Survey Lectures A Subseries of Proceedings of Symposia in Applied Mathematics Volume 40 MATRIX THEORY AND APPLICATIONS Edited by Charles R. Johnson (Phoenix, Arizona, January 1989) Volume 39 CHAOS AND FRACTALS: THE MATHEMATICS BEHIND THE COMPUTER GRAPHICS Edited by Robert L.. Devaney and Linda Keen (Providence, Rhode Island, August 1988) Volume 38 COMPUTATIONAL COMPLEXITY THEORY Edited by Juris Hartmanis (Atlanta, Georgia, January 1988) Volume 37 MOMENTS IN MATHEMATICS Edited by Henry J. Landau (San Antonio, Texas, January 1987) Volume 36 APPROXIMATION THEORY Edited by Carl de Boor (New Orleans, Louisiana, January 1986) Volume 35 ACTUARIAL MATHEMATICS Edited by Harry H. Panjer (Laramie, Wyoming, August 1985) Volume 34 MATHEMATICS OF INFORMATION PROCESSING Edited by Michael Anshel and William Gewirtz (Louisville, Kentucky, January 1984) Volume 33 FAIR ALLOCATION Edited by H. Peyton Young (Anaheim, California, January 1985) Volume 32 ENVIRONMENTAL AND NATURAL RESOURCE MATHEMATICS Edited by R. W. McKelvey (Eugene, Oregon, August 1984) Volume 31 COMPUTER COMMUNICATIONS Edited by B. Gopinath (Denver, Colorado, January 1983) Volume 30 POPULATION BIOLOGY Edited by Simon A. Levin (Albany, New York, August 1983) Volume 29 APPLIED CRYPTOLOGY, CRYPTOGRAPHIC PROTOCOLS, AND COMPUTER SECURITY MODELS By R. A. DeMillo, G. I. Davida, D. P. Dobkin, M. A. Harrison, and R. J. Lipton (San Francisco, California, January 1981) Volume 28 STATISTICAL DATA ANALYSIS Edited by R. Gnanadesikan (Toronto, Ontario, August 1982) Volume 27 COMPUTED TOMOGRAPHY Edited by L. A. Shepp (Cincinnati, Ohio, January 1982) Volume 26 THE MATHEMATICS OF NETWORKS Edited by S. A. Burr (Pittsburgh, Pennsylvania, August 1981) Volume 25 OPERATIONS RESEARCH: MATHEMATICS AND MODELS Edited by S.
    [Show full text]
  • Hadamard and Conference Matrices
    Journal of Algebraic Combinatorics 14 (2001), 103–117 c 2001 Kluwer Academic Publishers. Manufactured in The Netherlands. Hadamard and Conference Matrices K.T. ARASU Department of Mathematics and Statistics, Wright State University, Dayton, Ohio 45435, USA YU QING CHEN Department of Mathematics, RMIT-City Campus, GPO Box 2476V, Melbourne, Victoria 3001, Australia ALEXANDER POTT Institute for Algebra and Geometry, Otto-von-Guericke-University Magdeburg, 39016 Magdeburg, Germany Received October 11, 2000; Revised April 12, 2001 Abstract. We discuss new constructions of Hadamard and conference matrices using relative difference ( , , − , n−2 ) − sets. We present the first example of a relative n 2 n 1 2 -difference set where n 1 is not a prime power. Keywords: difference sets, relative difference sets, Hadamard matrices 1. Introduction One of the most interesting problems in combinatorics is the question whether Hadamard matrices exist for all orders n divisible by 4. A Hadamard matrix H of order n is a t ±1-matrix which satisfies HH = nIn. They exist for − − − 1 1 1 1 1 −1 −11−1 −1 n = 1, n = 2 , n = 4 −11 −1 −11−1 −1 −1 −11 and many more values of n. It is not difficult to see that a Hadamard matrix of order n satisfies n = 1, n = 2orn is divisible by 4. The smallest case n where it is presently not known whether a Hadamard matrix exists is n = 428. Many recursive or clever “ad hoc” constructions of Hadamard matrices are known; we refer the reader to [3, 5] and [23], for instance. We also refer the reader to [3] for background from design theory.
    [Show full text]
  • Complex Conference Matrices, Complex Hadamard Matrices And
    Complex conference matrices, complex Hadamard matrices and complex equiangular tight frames Boumediene Et-Taoui Universit´ede Haute Alsace – LMIA 4 rue des fr`eres Lumi`ere 68093 Mulhouse Cedex October 25, 2018 Abstract In this article we construct new, previously unknown parametric families of complex conference ma- trices and of complex Hadamard matrices of square orders and related them to complex equiangular tight frames. It is shown that for any odd integer k 3 such that 2k = pα + 1, p prime, α non-negative integer, on the one hand there exists a (2k, k) complex≥ equiangular tight frame and for any β N∗ there exists a 2β 1 2β−1 2β−1 ∈ ((2k) , 2 (2k) ((2k) 1)) complex equiangular tight frame depending on one unit complex num- ± 2β 1 2β−1 2β−1 ber, and on the other hand there exist a family of ((4k) , 2 (4k) ((4k) 1)) complex equiangular tight frames depending on two unit complex numbers. ± AMS Classification. Primacy 42C15, 52C17; Secondary 05B20. [email protected] 1 Introduction This article deals with the so-called notion of complex equiangular tight frames. Important for coding and quantum information theories are real and complex equiangular tight frames, for a survey we refer to [13]. In a Hilbert space , a subset F = f is called a frame for provided that there are two constants H { i}i I ⊂H H C,D > 0 such that ∈ C x 2 x,f 2 D x 2 , k k ≤ |h ii| ≤ k k i I X∈ holds for every x . If C = D = 1 then the frame is called normalized tight or a Parseval frame.
    [Show full text]
  • A Survey on Modular Hadamard Matrices ∗
    Morfismos, Vol. 7, No. 2, 2003, pp. 17{45 A survey on modular Hadamard matrices ∗ Shalom Eliahou Michel Kervaire Abstract We provide constructions of 32-modular Hadamard matrices for every size n divisible by 4. They are based on the description of several families of modular Golay pairs and quadruples. Higher moduli are also considered, such as 48, 64, 128 and 192. Finally, we exhibit infinite families of circulant modular Hadamard matrices of various types for suitable moduli and sizes. 2000 Mathematics Subject Classification: 05B20, 11L05, 94A99. Keywords and phrases: modular Hadamard matrix, modular Golay pair. 1 Introduction A square matrix H of size n, with all entries 1, is a Hadamard matrix if HHT = nI, where HT is the transpose of H and I the identity matrix of size n. It is easy to see that the order n of a Hadamard matrix must be 1, 2 or else a multiple of 4. There are two fundamental open problems about these matrices: • Hadamard's conjecture, according to which there should exist a Hadamard matrix of every size n divisible by 4. (See [9].) • Ryser's conjecture, stating that there probably exists no circulant Hadamard matrix of size greater than 4. (See [13].) Recall that a circulant matrix is a square matrix C = (ci;j)0≤i;j≤n−1 of size n, such that ci;j = c0;j−i for every i; j (with indices read mod n). ∗Invited article 17 18 Eliahou and Kervaire There are many known constructions of Hadamard matrices. How- ever, Hadamard's conjecture is widely open.
    [Show full text]
  • Graph Spectra and Continuous Quantum Walks Gabriel Coutinho
    Graph Spectra and Continuous Quantum Walks Gabriel Coutinho, Chris Godsil September 1, 2020 Preface If A is the adjacency matrix of a graph X, we define a transition matrix U(t) by U(t) = exp(itA). Physicists say that U(t) determines a continuous quantum walk. Most ques- tions they ask about these matrices concern the squared absolute values of the entries of U(t) (because these may be determined by measurement, and the entries themselves cannot be). The matrices U(t) are unitary and U(t) = U(−t), so we may say that the physicists are concerned with ques- tions about the entries of the Schur product M(t) = U(t) ◦ U(t) = U(t) ◦ U(−t). Since U(t) is unitary, the matrix M(t) is doubly stochastic. Hence each column determines a probability density on V (X) and there are two extreme cases: (a) A row of M(t) has one nonzero entry (necessarily equal to 1). (b) All entries in a row of U(t) are equal to 1/|V (X)|. In case (a), there are vertices a and b of X such that |U(t)a,b| = 1. If a =6 b, we have perfect state transfer from a to b at time t, otherwise we say that X is periodic at a at time t. Physicists are particularly interested in perfect state transfer. In case (b), if the entries of the a-row of M(t) is constant, we have local uniform mixing at a at time t. (Somewhat surprisingly these two concepts are connected—in a number of cases, perfect state transfer and local uniform mixing occur on the same graph.) The quest for graphs admitting perfect state transfer or uniform mixing has unveiled new results in spectral graph theory, and rich connections to other fields of mathematics.
    [Show full text]
  • Robust Hadamard Matrices and Equi-Entangled Bases
    Robust Hadamard matrices and equi-entangled bases Grzegorz Rajchel1;2 Coauthors of the paper: Adam G ˛asiorowski2, Karol Zyczkowski˙ 1;3 1Centrum Fizyki Teoretycznej PAN; 2 Wydział Fizyki, Uniwersytet Warszawski; 3Instytut Fizyki im. Smoluchowskiego, Uniwersytet Jagiellonski´ Introduction Now we can introduce another notion important for results of this work: Definition 8 A symmetric matrix with entries ±1 outside the diagonal and 0 at the diagonal, which satisfy or- Hadamard matrices with particular structure attract a lot of attention, as their existence is related to several prob- thogonality relations: T lems in combinatorics and mathematical physics. Usually one poses a question, whether for a given size n there CC = (n − 1)I; exists a Hadamard matrix with a certain symmetry or satisfying some additional conditions. is called a symmetric conference matrix. We introduce notion of robust Hadamard matrices. This will be useful to broaden our understanding of the problem of unistochasticity inside the Birkhoff polytope. Lemma 9 Matrices of the structure: R H = C + iI; Birkhoff polytope where C is symmetric conference matrix, are complex robust Hadamard matrices. Definition 1 The set Bn of bistochastic matrices (also called Birkhoff polytope) consists of matrices with non- negative entries which satisfy two sum conditions for rows and columns: Main result n n We proved following propositions: X X Bn = fB 2 Mn(R); Bij = Bji = 1; j = 1; : : : ; ng (1) Proposition 10 For every order n, for which there exists a symmetric conference matrix or a skew Hadamard i=1 i=1 matrix, every matrix belonging to any ray R or any counter-ray Re of the Birkhoff polytope Bn is unistochastic.
    [Show full text]