COMPLEX HADAMARD MATRICES AND APPLICATIONS TEO BANICA Abstract. A complex Hadamard matrix is a square matrix H 2 MN (C) whose entries are on the unit circle, jHijj = 1, and whose rows and pairwise orthogonal. The main ij 2πi=N example is the Fourier matrix, FN = (w ) with w = e . We discuss here the basic theory of such matrices, with emphasis on geometric and analytic aspects. CONTENTS Introduction 1 1. Hadamard matrices 5 2. Complex matrices 21 3. Roots of unity 37 4. Geometry, defect 53 5. Special matrices 69 6. Circulant matrices 85 7. Bistochastic matrices 101 8. Glow computations 117 9. Norm maximizers 133 10. Quantum groups 149 11. Subfactor theory 165 12. Fourier models 181 References 197 Introduction A complex Hadamard matrix is a square matrix H 2 MN (C) whose entries belong to the unit circle in the complex plane, Hij 2 T, and whose rows are pairwise orthogonal N P with respect to the usual scalar product of C , given by < x; y >= i xiy¯i. 2010 Mathematics Subject Classification. 15B10. Key words and phrases. Hadamard matrix, Fourier matrix. 1 2 TEO BANICA p The orthogonality condition tells us that the rescaled matrix U = H= N must be unitary. Thus, these matrices form a real algebraic manifold, given by: p XN = MN (T) \ NUN ij 2πi=N The basic example is the Fourier matrix, FN = (w ) with w = e . In standard matrix form, and with indices i; j = 0; 1;:::;N − 1, this matrix is as follows: 01 1 1 ::: 1 1 B1 w w2 : : : wN−1 C B C B1 w2 w4 : : : w2(N−1)C FN = B C B. C @. A 1 wN−1 w(2N−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 ' Gb as a square matrix, FG 2 MG(C): F =< i; j > G i2G;j2Gb Observe that for the cyclic group G = ZN we obtain in this way the above standard Fourier matrix FN . In general, we obtain a tensor product of Fourier matrices FN . There are many other examples of such matrices, for the most coming from various combinatorial constructions, basically involving design theory, and roots of unity. In addition, there are several deformation procedures for such matrices, leading to some more complicated constructions as well, or real algebraic geometry flavor. In general, the complex Hadamard matrices can be thought of as being \generalized Fourier matrices", of somewhat exotic type. Due to their generalized Fourier nature, these matrices appear in a wide array of questions in mathematics and physics: 1. Operator algebras. One important concept in the theory of von Neumann algebras is that of maximal abelian subalgebra (MASA). In the finite case, where the algebra has a trace, one can talk about pairs of orthogonal MASA. In the simplest case, of the matrix ∗ algebra MN (C), the orthogonal MASA are, up to conjugation, A = ∆;B = H∆H , where ∆ ⊂ MN (C) are the diagonal matrices, and H 2 MN (C) is Hadamard. 2. Subfactor theory. Along the same lines, but at a more advanced level, associated ∗ to any Hadamard matrix H 2 MN (C) is the square diagram C ⊂ ∆;H∆H ⊂ MN (C) formed by the associated MASA, which is a commuting square in the sense of subfactor theory. The Jones basic construction produces, out of this diagram, an index N subfactor of the Murray-von Neumann factor R, whose computation a key problem. HADAMARD MATRICES 3 3. Quantum groups. Associated to any complex Hadamard matrix H 2 MN (C) is a + certain quantum permutation group G ⊂ SN , obtained by factorizing the flat representa- + tion π : C(SN ) ! MN (C) associated to H. As a basic example here, the Fourier matrix FG produces in this way the group G itself. In general, the above-mentioned subfactor can be recovered from G, whose computation is a key problem. 4. Lattice models. According to the work of Jones, the combinatorics of the subfactor associated to an Hadamard matrix H 2 MN (C), which by the above can be recovered + from the representation theory of the associated quantum permutation group G ⊂ SN , can be thought of as being the combinatorics of a \spin model", in the context of link invariants, or of statistical mechanics, in an abstract, mathematical sense. From a more applied point of view, the Hadamard matrices can be used in order to construct mutually unbiased bases (MUB) and other useful objects, which can help in connection with quantum information theory, and other quantum physics questions. All this is quite recent, basically going back to the 00s. Regarding the known facts about 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 2 MN (±1), the study of such matrices goes back to the 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 matrix techniques apply quite well to the root of unity case, H 2 MN (Zs), with s < 1. 2. Geometry. As already explained above,p the N × N complex Hadamard matrices form a real algebraic manifold, XN = MN (T) \ NUN . This manifold is highly singular, but several interesting geometric results about it have been obtained, notably about the general structure of the singularity at a given point H 2 XN , about the neighborhood of the Fourier matrices FG, and about the various isolated points as well. 3. Analysis. One interesting point of view on the Hadamard matrices, realp or complex, comes from the fact that these are precisely the rescaled versions, H = NU, of the P matrices which maximize the 1-norm jjUjj1 = ij jUijj on ON ;UN respectively. When looking 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 playing a Gale-Berlekamp type game with the matrix, in the hope that the invariants which are obtained 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 this is not easy, and we have chosen an algebra/geometry/analysis/physics lineup for our presentation, 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 we had 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 was the 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 that I wrote or co-signed, mostly during 2005{2015, and I would like to thank my cowork- ers Julien Bichon, Beno^ıtCollins, Ion Nechita, Remus Nicoar˘a,Duygu Ozteke,¨ Lorenzo Pittau, 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 some beautiful pure mathematics, regarding the real case. The definition that we need, going back to 19th century work of Sylvester [85], is as follows: Definition 1.1. An Hadamard matrix is a square binary matrix, H 2 MN (±1) whose rows are pairwise orthogonal, with respect to the scalar product on RN . As a first observation, we do not really need real numbers in order to talk about the Hadamard matrices, because the orthogonality condition tells us that, when comparing two rows, the number of matchings should equal the number of mismatchings. Thus, we can 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 the foundations 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 p YN = MN (±1) \ NON where ON is the orthogonal group, the intersection being taken inside MN (R). p Proof. Let H 2 MN (±1). Since the rows of the rescaled matrix U = H= N have norm 1, with respect to the usual scalar product on RN , we conclude that H is Hadamard precisely when U belongs to the orthogonal group ON , and so when H 2 YN , as claimed.