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Extension of the Set of Complex Hadamard Matrices of Size 8 Math.Comput.Sci. (2018) 12:459–464 https://doi.org/10.1007/s11786-018-0379-8 Mathematics in Computer Science Extension of the Set of Complex Hadamard Matrices of Size 8 Wojciech T. Bruzda Received: 10 January 2018 / Revised: 5 February 2018 / Accepted: 20 March 2018 / Published online: 19 September 2018 © The Author(s) 2018 Abstract A complex Hadamard matrix is defined as a matrix H which fulfills two conditions, |H j,k|=1 for all ∗ j and k and HH = NIN where IN is an identity matrix of size N. We explore the set of complex Hadamard matrices HN of size N = 8 and present two previously unknown structures: a one-parametric, non-affine family (1) (0) T8 of complex Hadamard matrices and a single symmetric and isolated matrix A8 . Keywords Complex Hadamard matrix · Non-affine family · Isolated matrix · Butson class Mathematics Subject Classification 15A03 · 65F25 · 15B34 · 05B20 1 Introduction The description of the set HN of complex Hadamard matrices (CHMs) is exhausted for N ∈{2, 3, 4, 5} [1,2], while beginning from N = 6, general knowledge of the structure of HN can be considered incomplete. CHMs and their subset containing only real ∓1-entry objects are a subject of great interest in many fields of contemporary mathematics and physics. Let us only mention applications in quantum optics, telecommunication and signal processing, assistance in error correcting codes, background for equiangular lines, and many others [2–6]. Two matrices H1 and H2 ∈ HN are said to be equivalent if one of them can be expressed in terms of the other by means of two permutation matrices P1 and P2 and two unitary diagonal matrices D1 and D2 as H1 = P1 D1 H2 P2 D2 [2,7]. In general, the recognition of equivalence is not trivial and requires additional tools such as Haagerup invariants [2] or matrix fingerprint [8]. Another signature assigned to a matrix is its defect [7,9]. The defect of a unitary matrix d(H) tells us what is the largest possible size of a smooth family of CHMs stemming from H. This one-way criterion is useful when searching for isolated matrices for which the defect vanishes. There are two possible types of CHM families: affine and non-affine. In the first case, the particular constant entries in given matrix H can be extended to complex variables so that their phases vary linearly as in the following example of the 4-dimensional Fourier matrix [1] W. T. Bruzda (B) Institute of Physics, Jagiellonian University, Kraków, Poland e-mail: [email protected] 460 W. T. Bruzda ⎡ ⎤ ⎡ ⎤ 1111 1111 ⎢ 1 i −1 −i ⎥ ⎢ 1 ieiα −1 −ieiα ⎥ F = ⎢ ⎥ → ⎢ ⎥ : α ∈[0, 2π). (1.1) 4 ⎣ 1 −11−1 ⎦ ⎣ 1 −11 −1 ⎦ 1 −i −1 i 1 −ieiα −1 ieiα The second case involves non-linear formulas for the phases of the variable entries [7,12]. Every CHM can be transformed by an equivalence preserving operation into the form in which its first row and its first column has only ones. The remaining (N − 1) × (N − 1) elements shall be called the core of the matrix. All matrices in this paper are worked out in the above dephased form. A special subclass of CHMs is called Butson type matrices [10,11], denoted by BH(N, q) ⊂ HN , with the q property that each entry of H ∈ BH(N, q) is the qth root of unity, H j,k = 1 for any j and k. For instance, the standard class of real Hadamard matrices of size N forms the class BH(N, 2), while the Fourier matrix FN of order N belongs to the set BH(N, N). The notation adopted in this work is entirely taken from the on-line CHM catalogue [12] based on [7] and, to save space, we use the symbol [β] as a shortcut for the exponent expression exp(iπβ). 2 Looking for a New Matrix H ∈ H8 Searching for new CHMs supports the description of the set HN and provides better understanding of its elements. Moreover, additional profit from having new examples of CHMs is extension√ of the area for searching for new ∗ / ∈ H mutually unbiased pairs of Hadamard matrices H1 and H2 such that H1 H2 N N , which is directly related to mutually unbiased bases [13–15]. Looking for possibly new complex Hadamard matrices, the following procedure was used. Let us take any dephased matrix H with unimodular entries H j,k =[ϕ j,k] for ϕ j,k ∈[0, 2). Define a target function Z(H) ≡ ∗ − I 2 = ∗ HH N N F, where the Frobenius norm of matrix is applied, X F trXX . Using a random-walk procedure over the core phases we try to minimize Z(H) and for Z(H)<10−16 we can stop having a matrix H that belongs to HN with respect to a given precision. Such raw matrices are ready for further analytical processing. The most interesting outcomes show up when only a subset of core phases undergoes variation while the other elements are fixed to certain values. The algorithm is very fast and even using an average desktop PC for N < 9 one can obtain results within seconds. The case of N = 6, however without any formal proof of completeness, seems to be already understood and it (4) (3) is conjectured that the families G6 of Szöll˝osi [16] and K6 of Karlsson [17] along with the isolated Tao matrix (0) ∈ H H S6 [18,19] constitute the set in which any H 6 is included. The set 7 consists of several isolated matrices: (0) (0) (0) (1) F7 [7], C7 [2,20] and Q7 [21] together with one affine family P7 [22]. The investigation of this case is far from being closed but proved to be hard for any numerical attempts. Hence, in this short paper, we focus on the next promising case for N = 8 and, as a consequence, we present previously unknown: • (1) ( , ) a 1-parametric, non-affine family T8 of CHMs that contains Butson BH 8 20 matrices, • (0) an isolated matrix A8 of a non-Butson type. (1) = 3 Non-affine Family T8 of N 8 Complex Hadamard Matrices Let us start with the matrix T8 ∈ H8 that was found numerically (see Sect. 2) and then expressed in the analytic form using symmetric patterns in the elements and phases of the core and orthogonality constraints with respect to the rows and columns. Define a constant matrix: Extension of the Set of Complex Hadamard Matrices of Size 8 461 ⎛ ⎡ ⎤⎞ •••••••• ⎜ ⎢ ⎥⎟ ⎜ ⎢ ••• • • 31518⎥⎟ ⎜ ⎢ ⎥⎟ ⎜ ⎢ ••• • •13 15 8 ⎥⎟ ⎜ ⎢ ⎥⎟ ⎜ π ⎢ • 22 2 2175 2⎥⎟ T = EXP ⎜i ⎢ ⎥⎟ (3.1) ⎜ 10 ⎢ • 22227512⎥⎟ ⎜ ⎢ ⎥⎟ ⎜ ⎢ ••• • • •10 10 ⎥⎟ ⎝ ⎣ ⎦⎠ • 22121210 • 10 • 2212121010 • where T = EXP(A) denotes the entrywise exponent of a matrix, Tj,k = exp(A j,k) and • replaces zero. Furthermore, we shall use the matrix ⎡ ⎤ 1 1 1 1 1111 ⎢ − / ⎥ ⎢ 1 c3 c2 c1 i c1 111⎥ ⎢ ⎥ ⎢ 1 −c1 i/c1 −i/c2 −i/c3 111⎥ ⎢ ⎥ ⎢ 1 c2 c3 −i/c1 c1 111⎥ Tc = ⎢ ⎥ . (3.2) ⎢ 1 −i/c1 c1 i/c3 i/c2 111⎥ ⎢ α α − α − α ⎥ ⎢ 1 ei −ei −e i e i 111⎥ ⎣ α α − α − α ⎦ 1 ei −ei −e i e i 111 1 1 1 1 1111 with 1 √ 1 √ c1 = 8 − 50 − 6 5 + i 8 + 50 − 6 5, (3.3) 4 4 3 4 i − 1 − + − c1 + i/c1 c = 10 5 , (3.4) 2 − 2 1 ic1 =− 2, c3 ic2c1 (3.5) √ where α =−arccot 1 − 5 and [β] = exp(iπβ). This allows us to define the following complex Hadamard matrix T8 = Tc ◦ T where ◦ is the entrywise product of matrices. One can check that the defect d(T8) = 3 which is an indication that the matrix can be a part of a family of inequivalent CHMs. Moreover, the structure of the three last columns resembles the form of a Butson type object which might suggest that a Butson matrix lies on a trajectory of the orbit stemming from a given matrix and some entries (T8) j,k for 2 j 7 and 2 k 5 could be functionally dependent. Indeed, when one assigns to two particular elements u(μ) = (T8)7,2 and −u(μ) = (T8)7,3 a value μ ∈[0, 1/2] for u =[μ], having the last three columns fixed, there always can be found a CHM with the second row of the form yz 3 18 1, x, y, z, − , , −i, . (3.6) x 10 10 462 W. T. Bruzda Again, using symmetries amongst phases and orthogonality constraints, one recovers a full matrix in the form of ⎛ ⎡ ⎤⎞ ⎡ ⎤ ••••• • • • 11111111 ⎜ ⎢ ⎥⎟ ⎢ 1 xy z− yz 111⎥ ⎜ ⎢ ••••• 31518⎥⎟ ⎢ x ⎥ ⎜ ⎢ ⎥⎟ ⎢ − u u − uz − uz ⎥ • 888813158 ⎢ 1 2 111⎥ ⎜ ⎢ ⎥⎟ ⎢ y x xy x ⎥ ⎜ π ⎢ • ⎥⎟ ⎢ 1 yx− yz z 111⎥ (1)( ) = ⎜ ⎢ 2222175 2⎥⎟ ◦ ⎢ x ⎥ T8 x EXP ⎜i ⎢ ••••• ⎥⎟ ⎢ 1 u − u − uz − uz 111⎥ ⎜ 10 ⎢ 7512⎥⎟ ⎢ x y x2 xy ⎥ ⎜ ⎢ ••••• • ⎥⎟ ⎢ − x x − z z ⎥ ⎜ ⎢ 10 10 ⎥⎟ ⎢ 1 i z i z i x i x 111⎥ ⎝ ⎣ ⎦⎠ 2 2 •••••10 • 10 ⎣ u −u − uz uz ⎦ 1 x2 x2 111 •••••10 10 • uz uz − uz − uz 1 i x i x i x i x 111 (3.7) where √ √ 1 χ1 = 5 − 3 5 + 4 5 − 2 5 + (1 + i) + ζ2, (3.8) x √ √ χ χ = 45 − 9i − (17 − 7i) 5 − 2 620 − 500i − (285 − 234i) 5 − (1 + i) 1 , (3.9) 2 x √ √ χ2 χ3 = 10i − 10 + (6 − 2i) 5 + 4 10 − 10i − (5 − 4i) 5 + (1 + i) , (3.10) x √ √ √ χ3 χ4 = 5 − 1 + i 2 5 + 5 + (1 + i) , (3.11) x √ √ √ 2 ∗ χ5 = (1 + i) 1 − 5 + ζ4 − (1 − i) 2x + ( χ4) − x(1 − 4i + 5 + ζ1), (3.12) √ √ χ6 = 5 − 1 − i − 5 − 2 5 + (1 − i)x, (3.13) 1 χ y = 5 , χ (3.14) 4 6 x 4 z = (1 + i) − x − y − 1 + i , (3.15) x − y 5 √ √ 4 with ζ1 =−i 10 − 2 5, ζ2 =−ζ1 and ζ4 = i 10 + 2 5 being the roots of the polynomial equation ζ + 20ζ 2 + 80 = 0 and eventually i x2 − zx + xy + yz u = .
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