A complex Hadamard matrix is a square matrix H ∈ M N (C) whose entries are on the unit circle, |H ij | = 1, and whose rows and pairwise orthogonal. The main example is the Fourier matrix, F N = (w ij) with w = e 2πi/N. We discuss here the basic
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  COMPLEX HADAMARD MATRICES AND APPLICATIONS TEO BANICA Abstract.  A complex Hadamard matrix is a square matrix  H   ∈  M  N  ( C ) whose entriesare on the unit circle,  | H  ij |  = 1, and whose rows and pairwise orthogonal. The mainexample is the Fourier matrix,  F  N   = ( w ij ) with  w  =  e 2 πi/N  . We discuss here the basictheory of such matrices, with emphasis on geometric and analytic aspects. C ONTENTS Introduction 11. Hadamard matrices 52. Complex matrices 213. Roots of unity 374. Geometry, defect 535. Special matrices 696. Circulant matrices 857. Bistochastic matrices 1018. Glow computations 1179. Norm maximizers 13310. Quantum groups 14911. Subfactor theory 16512. Fourier models 181References 197 Introduction A complex Hadamard matrix is a square matrix  H   ∈  M  N  ( C ) whose entries belong tothe unit circle in the complex plane,  H  ij  ∈  T , and whose rows are pairwise orthogonalwith respect to the usual scalar product of   C N  , given by  < x,y > =  i x i ¯ y i . 2010  Mathematics Subject Classification.  15B10. Key words and phrases.  Hadamard matrix, Fourier matrix. 1  2 TEO BANICA The orthogonality condition tells us that the rescaled matrix  U   =  H/ √  N   must beunitary. Thus, these matrices form a real algebraic manifold, given by: X  N   =  M  N  ( T ) ∩√  NU  N  The basic example is the Fourier matrix,  F  N   = ( w ij ) with  w  =  e 2 πi/N  . In standardmatrix form, and with indices  i,j  = 0 , 1 ,...,N   − 1, this matrix is as follows: F  N   =  1 1 1  ...  11  w w 2 ... w N  − 1 1  w 2 w 4 ... w 2( N  − 1) ... ... ... ...1  w N  − 1 w (2 N  − 1) ... w ( N  − 1) 2  More generally, we have as example the Fourier coupling of any finite abelian group  G ,regarded via the isomorphism  G  ≃   G  as a square matrix,  F  G  ∈  M  G ( C ): F  G  = < i,j > i ∈ G,j ∈   G Observe that for the cyclic group  G  =  Z N   we obtain in this way the above standardFourier matrix  F  N  . In general, we obtain a tensor product of Fourier matrices  F  N  .There are many other examples of such matrices, for the most coming from variouscombinatorial constructions, basically involving design theory, and roots of unity. Inaddition, there are several deformation procedures for such matrices, leading to somemore complicated constructions as well, or real algebraic geometry flavor.In general, the complex Hadamard matrices can be thought of as being “generalizedFourier matrices”, of somewhat exotic type. Due to their generalized Fourier nature, thesematrices appear in a wide array of questions in mathematics and physics: 1. Operator algebras.  One important concept in the theory of von Neumann algebrasis that of maximal abelian subalgebra (MASA). In the finite case, where the algebra hasa trace, one can talk about pairs of orthogonal MASA. In the simplest case, of the matrixalgebra  M  N  ( C ), the orthogonal MASA are, up to conjugation,  A  = ∆ ,B  =  H  ∆ H  ∗ , where∆  ⊂  M  N  ( C ) are the diagonal matrices, and  H   ∈  M  N  ( C ) is Hadamard. 2. Subfactor theory.  Along the same lines, but at a more advanced level, associatedto any Hadamard matrix  H   ∈  M  N  ( C ) is the square diagram  C  ⊂  ∆ ,H  ∆ H  ∗  ⊂  M  N  ( C )formed by the associated MASA, which is a commuting square in the sense of subfactortheory. The Jones basic construction produces, out of this diagram, an index  N   subfactorof the Murray-von Neumann factor  R , whose computation a key problem.  HADAMARD MATRICES 3 3. Quantum groups.  Associated to any complex Hadamard matrix  H   ∈  M  N  ( C ) is acertain quantum permutation group  G  ⊂  S  + N  , obtained by factorizing the flat representa-tion  π  :  C  ( S  + N  )  →  M  N  ( C ) associated to  H  . As a basic example here, the Fourier matrix F  G  produces in this way the group  G  itself. In general, the above-mentioned subfactorcan be recovered from  G , whose computation is a key problem. 4. Lattice models.  According to the work of Jones, the combinatorics of the subfactorassociated to an Hadamard matrix  H   ∈  M  N  ( C ), which by the above can be recoveredfrom the representation theory of the associated quantum permutation group  G  ⊂  S  + N  ,can be thought of as being the combinatorics of a “spin model”, in the context of linkinvariants, or of statistical mechanics, in an abstract, mathematical sense.From a more applied point of view, the Hadamard matrices can be used in order toconstruct mutually unbiased bases (MUB) and other useful objects, which can help inconnection with quantum information theory, and other quantum physics questions.All this is quite recent, basically going back to the 00s. Regarding the known factsabout the Hadamard matrices, most of them are in fact of purely mathematical nature.There are indeed many techniques that can be applied, leading to various results: 1. Algebra.  In the real case,  H   ∈  M  N  ( ± 1), the study of such matrices goes back tothe beginning of the 20th century, and is quite advanced. The main problems, however,namely the Hadamard conjecture (HC) and the circulant Hadamard conjecture (CHC)are not solved yet, with no efficient idea of approach in sight. Part of the real matrixtechniques apply quite well to the root of unity case,  H   ∈  M  N  ( Z s ), with  s <  ∞ . 2. Geometry.  As already explained above, the  N   × N   complex Hadamard matricesform a real algebraic manifold,  X  N   =  M  N  ( T ) ∩√  NU  N  . This manifold is highly singular,but several interesting geometric results about it have been obtained, notably about thegeneral structure of the singularity at a given point  H   ∈  X  N  , about the neighborhood of the Fourier matrices  F  G , and about the various isolated points as well. 3. Analysis.  One interesting point of view on the Hadamard matrices, real or complex,comes from the fact that these are precisely the rescaled versions,  H   = √  NU  , of thematrices which maximize the 1-norm  || U  || 1  =   ij | U  ij |  on  O N  ,U  N   respectively. Whenlooking more generally at the local maximizers of the 1-norm, one is led into a notion of “almost Hadamard matrices”, having interesting algebraic and analytic aspects. 4. Probability.  Another speculative approach, this time probabilistic, is by playinga Gale-Berlekamp type game with the matrix, in the hope that the invariants which areobtained in this way are related to the various geometric and quantum algebraic invariants,  4 TEO BANICA which are hard to compute. All this is related to the subtle fact that any unitary matrix,and so any complex Hadamard matrix as well, can be put in bistochastic form.Our aim here is to survey this material, theory and applications. Organizing all thisis not easy, and we have chosen an algebra/geometry/analysis/physics lineup for ourpresentation, vaguely coming from the amount of background which is needed.The present text is organized in 4 parts, as follows:(1) Sections 1-3 contain basic definitions and various algebraic results.(2) Sections 4-6 deal with differential and algebraic geometric aspects.(3) Sections 7-9 are concerned with various analytic considerations.(4) Sections 10-12 deal with various mathematical physics aspects.There are of course many aspects of the theory which are missing from our presentation,but we will provide of course some information here, comments and references. Acknowledgements. I would like to thank Vaughan Jones for suggesting me, back to a discussion that wehad in 1997, when we first met, to look at vertex models, and related topics.Stepping into bare Hadamard matrices is quite an experience, and very inspiring wasthe work of Uffe Haagerup on the subject, and his papers [32], [51], [52].The present text is heavily based on a number of research papers on the subject thatI wrote or co-signed, mostly during 2005–2015, and I would like to thank my cowork-ers Julien Bichon, Benoˆıt Collins, Ion Nechita, Remus Nicoar˘a, Duygu ¨Ozteke, LorenzoPittau, Jean-Marc Schlenker, Adam Skalski and Karol ˙Zyczkowski.Finally, many thanks go to my cats, for advice with hunting techniques, martial arts,and more. When doing linear algebra, all this knowledge is very useful.  HADAMARD MATRICES 5 1.  Hadamard matrices We are interested here in the complex Hadamard matrices, but we will start with somebeautiful pure mathematics, regarding the real case. The definition that we need, goingback to 19th century work of Sylvester [85], is as follows: Definition 1.1.  An Hadamard matrix is a square binary matrix, H   ∈  M  N  ( ± 1) whose rows are pairwise orthogonal, with respect to the scalar product on   R N  . As a first observation, we do not really need real numbers in order to talk about theHadamard matrices, because the orthogonality condition tells us that, when comparingtwo rows, the number of matchings should equal the number of mismatchings. Thus, wecan replace if we want the 1 , − 1 entries of our matrix by any two symbols, of our choice.Here is an example of an Hadamard matrix, with this convention: ♥ ♥ ♥ ♥♥ ♣ ♥ ♣♥ ♥ ♣ ♣♥ ♣ ♣ ♥ However, it is probably better to run away from this, and use real numbers instead,as in Definition 1.1, with the idea in mind of connecting the Hadamard matrices to thefoundations of modern mathematics, namely Calculus 1 and Calculus 2.So, getting back now to the real numbers, here is a first result: Proposition 1.2.  The set of the   N   × N   Hadamard matrices is  Y  N   =  M  N  ( ± 1) ∩√  NO N  where   O N   is the orthogonal group, the intersection being taken inside   M  N  ( R ) .Proof.  Let  H   ∈  M  N  ( ± 1). Since the rows of the rescaled matrix  U   =  H/ √  N   have norm 1,with respect to the usual scalar product on R N  , we conclude that  H   is Hadamard preciselywhen  U   belongs to the orthogonal group  O N  , and so when  H   ∈  Y  N  , as claimed.   As an interesting consequence of the above result, which is not exactly obvious whenusing the design theory approach, we have the following result: Proposition 1.3.  Let   H   ∈  M  N  ( ± 1)  be an Hadamard matrix. (1)  The columns of   H   must be pairwise orthogonal. (2)  The transpose matrix   H  t ∈  M  N  ( ± 1)  is Hadamard as well.Proof.  Since the orthogonal group  O N   is stable under transposition, so is the set  Y  N  constructed in Proposition 1.2, and this gives both the assertions.  
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