# American Institute of Mathematical Sciences

December  2019, 12(8): 2349-2364. doi: 10.3934/dcdss.2019147

## Informing the structure of complex Hadamard matrix spaces using a flow

 1 Department of Mathematics, Duke University, Box 90320, Durham, NC 27708-0320, USA 2 Colorado State University, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA

Received  July 2016 Revised  December 2016 Published  January 2019

A complex Hadamard matrix $H$ may be isolated or may lie in a higher-dimensional space of Hadamards. We provide an upper bound for this dimension as the dimension of the center subspace of a gradient flow and apply the Center Manifold Theorem of dynamical systems theory to study local structure in spaces of complex Hadamard matrices. Through examples, we provide several applications of our methodology including the construction of affine families of Hadamard matrices.

Citation: Francis C. Motta, Patrick D. Shipman. Informing the structure of complex Hadamard matrix spaces using a flow. Discrete & Continuous Dynamical Systems - S, 2019, 12 (8) : 2349-2364. doi: 10.3934/dcdss.2019147
##### References:
 [1] A. A. Agaian, Hadamard Matrices and their Applications, Springer-Verlag, Berlin, 1985. doi: 10.1007/BFb0101073.  Google Scholar [2] N. Barros e Sá and I. Bengtsson, Families of complex Hadamard matrices, Lin. Alg. Appl., 438 (2013), 2929-2957.  doi: 10.1016/j.laa.2012.10.029.  Google Scholar [3] K. Beauchamp and R. Nicoara, Orthogonal maximal abelian *-subalgebras of the 6 × 6 matrices, Lin. Alg. Appl., 428 (2008), 1833-1853.  doi: 10.1016/j.laa.2007.10.023.  Google Scholar [4] R. Craigen, Equivalence Classes of Inverse Orthogonal and Unit Hadamard, Bull. Austral. Math. Soc., 44 (1991), 109-115.  doi: 10.1017/S0004972700029506.  Google Scholar [5] P. Diţǎ, Some results on the parametrization of complex Hadamard matrices, J. Phys. A, 20 (2004), 5355-5374.  doi: 10.1088/0305-4470/37/20/008.  Google Scholar [6] D. Goyeneche, A new method to construct families of complex Hadamard matrices in even dimensions, J. Math. Phys., 54 (2013), 032201, 18pp.  doi: 10.1063/1.4794068.  Google Scholar [7] U. Haagerup, Orthogonal maximal abelian *-subalgebras of the $n\times n$ matrices and cyclic n-roots, Operator Algebras and Quantum Field Theory (Rome), Cambridge, MA International Press, (1997), 296-322.  Google Scholar [8] J. Hadamard, Resolution d'une question relative aux determinants, Bull. des Sci. Math., 17 (1893), 240-246.   Google Scholar [9] A. S. Hedayat, N. J. A. Sloane and J. Stufken, Orthogonal Arrays, Springer Series in Statistics, New York, Springer, 1999. doi: 10.1007/978-1-4612-1478-6.  Google Scholar [10] I. Jex, S. Stenholm and A. Zeilinger, Hamiltonian theory of a symmetric multiport, Opt. Commun., 117 (1995), 95-101.  doi: 10.1016/0030-4018(95)00078-M.  Google Scholar [11] B. R. Karlsson, BCCB complex Hadamard matrices of order 9, and MUBs, Lin. Alg. Appl., 504 (2016), 309-324.  doi: 10.1016/j.laa.2016.04.012.  Google Scholar [12] B. R. Karlsson, Two-parameter complex Hadamard matrices for N = 6, J. Math. Phys., 50 (2009), 082104, 8pp.  doi: 10.1063/1.3198230.  Google Scholar [13] B. R. Karlsson, Three-parameter complex Hadamard matrices of order 6, Lin. Alg. Appl., 434 (2011), 247-258.  doi: 10.1016/j.laa.2010.08.020.  Google Scholar [14] P. H. J. Lampio, F. Szöllősi and P. R. J. Östergård, The quaternary complex Hadamard matrices of orders 10, 12, and 14, Discrete Mathematics, 313 (2013), 189-206.  doi: 10.1016/j.disc.2012.10.001.  Google Scholar [15] T. K. Leen, A coordinate-independent center manifold reduction, Phys. Lett. A, 174 (1993), 89-93.  doi: 10.1016/0375-9601(93)90548-E.  Google Scholar [16] D. W. Leung, Simulation and reversal of n-qubit Hamiltonians using Hadamard matrices, J. Mod. Opt., 49 (2002), 1199-1217.  doi: 10.1080/09500340110109674.  Google Scholar [17] M. Matolcsi, J. Réffy and F. Szöllősi, Constructions of complex Hadamard matrices via tiling abelian groups, Open Syst. Inf. Dyn., 14 (2007), 247-263.  doi: 10.1007/s11080-007-9050-6.  Google Scholar [18] D. McNulty and S. Weigert, Isolated Hadamard matrices from mutually unbiased product bases, J. Math. Phys., 53 (2012), 122202, 16pp..  doi: 10.1063/1.4764884.  Google Scholar [19] J. Meiss, Differential Dynamical Systems, SIAM, (2007).  doi: 10.1137/1.9780898718232.  Google Scholar [20] M. Reck, A. Zeilinger, H. J. Bernstein and P. Bertani, Experimental realization of any discrete unitary operator, Phys. Rev. Lett., 73 (1994), 58-61.  doi: 10.1103/PhysRevLett.73.58.  Google Scholar [21] F. Szöllősi and M. Matolcsi, Towards a classification of 6 × 6 complex Hadamard matrices, Open Syst. Inf. Dyn., 15 (2008), 93-108.  doi: 10.1142/S1230161208000092.  Google Scholar [22] F. Szöllősi, Complex Hadamard matrices of order 6: a four-parameter family, J. London Math. Soc., 85 (2012), 616-32.  doi: 10.1112/jlms/jdr052.  Google Scholar [23] F. Szöllősi, Parametrizing complex Hadamard matrices, European J. Combin., 29 (2008), 1219-1234.  doi: 10.1016/j.ejc.2007.06.009.  Google Scholar [24] W. Tadej and K. Życzkowski, A concise guide to complex Hadamard matrices, Open Syst. Inform. Dyn., 13 (2006), 133-177.  doi: 10.1007/s11080-006-8220-2.  Google Scholar [25] W. Tadej and K. Życzkowski, Defect of a unitary matrix, Lin. Alg. Appl., 429 (2008), 447-481.  doi: 10.1016/j.laa.2008.02.036.  Google Scholar [26] F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Springer-Verlag, Berlin, 1990. doi: 10.1007/978-3-642-97149-5.  Google Scholar [27] R. F. Werner, All teleportation and dense coding schemes, J. Phys. A: Math. Gen., 34 (2001), 7081-7094.  doi: 10.1088/0305-4470/34/35/332.  Google Scholar

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##### References:
 [1] A. A. Agaian, Hadamard Matrices and their Applications, Springer-Verlag, Berlin, 1985. doi: 10.1007/BFb0101073.  Google Scholar [2] N. Barros e Sá and I. Bengtsson, Families of complex Hadamard matrices, Lin. Alg. Appl., 438 (2013), 2929-2957.  doi: 10.1016/j.laa.2012.10.029.  Google Scholar [3] K. Beauchamp and R. Nicoara, Orthogonal maximal abelian *-subalgebras of the 6 × 6 matrices, Lin. Alg. Appl., 428 (2008), 1833-1853.  doi: 10.1016/j.laa.2007.10.023.  Google Scholar [4] R. Craigen, Equivalence Classes of Inverse Orthogonal and Unit Hadamard, Bull. Austral. Math. Soc., 44 (1991), 109-115.  doi: 10.1017/S0004972700029506.  Google Scholar [5] P. Diţǎ, Some results on the parametrization of complex Hadamard matrices, J. Phys. A, 20 (2004), 5355-5374.  doi: 10.1088/0305-4470/37/20/008.  Google Scholar [6] D. Goyeneche, A new method to construct families of complex Hadamard matrices in even dimensions, J. Math. Phys., 54 (2013), 032201, 18pp.  doi: 10.1063/1.4794068.  Google Scholar [7] U. Haagerup, Orthogonal maximal abelian *-subalgebras of the $n\times n$ matrices and cyclic n-roots, Operator Algebras and Quantum Field Theory (Rome), Cambridge, MA International Press, (1997), 296-322.  Google Scholar [8] J. Hadamard, Resolution d'une question relative aux determinants, Bull. des Sci. Math., 17 (1893), 240-246.   Google Scholar [9] A. S. Hedayat, N. J. A. Sloane and J. Stufken, Orthogonal Arrays, Springer Series in Statistics, New York, Springer, 1999. doi: 10.1007/978-1-4612-1478-6.  Google Scholar [10] I. Jex, S. Stenholm and A. Zeilinger, Hamiltonian theory of a symmetric multiport, Opt. Commun., 117 (1995), 95-101.  doi: 10.1016/0030-4018(95)00078-M.  Google Scholar [11] B. R. Karlsson, BCCB complex Hadamard matrices of order 9, and MUBs, Lin. Alg. Appl., 504 (2016), 309-324.  doi: 10.1016/j.laa.2016.04.012.  Google Scholar [12] B. R. Karlsson, Two-parameter complex Hadamard matrices for N = 6, J. Math. Phys., 50 (2009), 082104, 8pp.  doi: 10.1063/1.3198230.  Google Scholar [13] B. R. Karlsson, Three-parameter complex Hadamard matrices of order 6, Lin. Alg. Appl., 434 (2011), 247-258.  doi: 10.1016/j.laa.2010.08.020.  Google Scholar [14] P. H. J. Lampio, F. Szöllősi and P. R. J. Östergård, The quaternary complex Hadamard matrices of orders 10, 12, and 14, Discrete Mathematics, 313 (2013), 189-206.  doi: 10.1016/j.disc.2012.10.001.  Google Scholar [15] T. K. Leen, A coordinate-independent center manifold reduction, Phys. Lett. A, 174 (1993), 89-93.  doi: 10.1016/0375-9601(93)90548-E.  Google Scholar [16] D. W. Leung, Simulation and reversal of n-qubit Hamiltonians using Hadamard matrices, J. Mod. Opt., 49 (2002), 1199-1217.  doi: 10.1080/09500340110109674.  Google Scholar [17] M. Matolcsi, J. Réffy and F. Szöllősi, Constructions of complex Hadamard matrices via tiling abelian groups, Open Syst. Inf. Dyn., 14 (2007), 247-263.  doi: 10.1007/s11080-007-9050-6.  Google Scholar [18] D. McNulty and S. Weigert, Isolated Hadamard matrices from mutually unbiased product bases, J. Math. Phys., 53 (2012), 122202, 16pp..  doi: 10.1063/1.4764884.  Google Scholar [19] J. Meiss, Differential Dynamical Systems, SIAM, (2007).  doi: 10.1137/1.9780898718232.  Google Scholar [20] M. Reck, A. Zeilinger, H. J. Bernstein and P. Bertani, Experimental realization of any discrete unitary operator, Phys. Rev. Lett., 73 (1994), 58-61.  doi: 10.1103/PhysRevLett.73.58.  Google Scholar [21] F. Szöllősi and M. Matolcsi, Towards a classification of 6 × 6 complex Hadamard matrices, Open Syst. Inf. Dyn., 15 (2008), 93-108.  doi: 10.1142/S1230161208000092.  Google Scholar [22] F. Szöllősi, Complex Hadamard matrices of order 6: a four-parameter family, J. London Math. Soc., 85 (2012), 616-32.  doi: 10.1112/jlms/jdr052.  Google Scholar [23] F. Szöllősi, Parametrizing complex Hadamard matrices, European J. Combin., 29 (2008), 1219-1234.  doi: 10.1016/j.ejc.2007.06.009.  Google Scholar [24] W. Tadej and K. Życzkowski, A concise guide to complex Hadamard matrices, Open Syst. Inform. Dyn., 13 (2006), 133-177.  doi: 10.1007/s11080-006-8220-2.  Google Scholar [25] W. Tadej and K. Życzkowski, Defect of a unitary matrix, Lin. Alg. Appl., 429 (2008), 447-481.  doi: 10.1016/j.laa.2008.02.036.  Google Scholar [26] F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Springer-Verlag, Berlin, 1990. doi: 10.1007/978-3-642-97149-5.  Google Scholar [27] R. F. Werner, All teleportation and dense coding schemes, J. Phys. A: Math. Gen., 34 (2001), 7081-7094.  doi: 10.1088/0305-4470/34/35/332.  Google Scholar
Plot of the eigenvalues of the linearization of $\Phi_4$ at ${\boldsymbol \theta}(a)$ for $a \in [0,\pi]$. $\lambda_1(a)$ (blue), $\lambda_3(a)$ (green), and $\lambda_6(a)$ (red) simultaneously vanish at $a = \pi/2$, while all other eigenvalues (gray) are strictly negative for all $a \in [0,\pi]$
Snapshots of 500 initial phases, drawn from $\mathbb{R}^{9}$ uniformly at random from a neighborhood of the core phases corresponding to $F(\pi/2)$, as they evolve under the flow defined by $\Phi_4({\boldsymbol \theta})$, at times (ⅰ) 5, (ⅱ) 20, (ⅲ) 70, and (ⅳ) 500. Each point cloud has been projected onto its top three principal components, and each point ${\boldsymbol \theta}$ is colored by $\log_{10}$ of the magnitude of the vector field $\Phi_4({\boldsymbol \theta})$
Plot of the 25 eigenvalues of $D\Phi_6\vert_{D_6(c)}$ for $c \in [-\pi/2,\pi/2]$. The zero eigenvalue (blue) has multiplicity four, the roots of $f_1(\lambda;c)$ (green) have multiplicity two, and all other eigenvalues (gray) are simple
Snapshots of 500 initial phases, drawn from $\mathbb{R}^{64}$ uniformly at random from a neighborhood of the core phases corresponding to $B_9^{(0)}$, as they evolve under the flow defined by $\Phi_9({\boldsymbol \theta})$, at times (ⅰ) 5, (ⅱ) 20, (ⅲ) 70, and (ⅳ, ⅴ) 500. Each point cloud has been projected onto its top three principal components, and each point ${\boldsymbol \theta}$ is colored by $\log_{10}$ of the magnitude of the vector field $\Phi_9({\boldsymbol \theta})$
Nonzero coordinates of the vectors in the basis $\{\textbf{V}_1, \ldots, \textbf{V}_{16}\}$ for $D\Phi_{10}\vert_{D_{10}}$. A nonzero coordinate has value 1 or -1, indicated by the subcolumn to which it belongs
 1 -1 vector coordinate $\textbf{V}_1$ 2 3 7 8 74 75 79 80 10 18 19 27 55 63 64 72 $\textbf{V}_2$ 10 12 16 18 64 66 70 72 2 8 20 26 56 62 74 80 $\textbf{V}_3$ 28 29 35 36 46 47 53 54 4 6 13 15 67 69 76 78 $\textbf{V}_4$ 4 8 24 25 40 44 78 79 28 32 48 54 57 63 64 68 $\textbf{V}_5$ 37 40 47 54 55 58 65 72 5 7 15 17 32 34 78 80 $\textbf{V}_6$ 4 9 12 14 48 50 67 72 20 24 28 35 38 42 73 80 $\textbf{V}_7$ 19 27 29 34 38 43 64 72 3 8 13 14 58 59 75 80 $\textbf{V}_8$ 37 39 43 45 46 48 52 54 5 6 23 24 59 60 77 78 $\textbf{V}_9$ 2 8 20 26 43 45 52 54 10 12 59 60 64 66 77 78 $\textbf{V}_{10}$ 47 48 49 50 74 75 76 77 15 18 24 27 33 36 42 45 $\textbf{V}_{11}$ 2 6 30 32 65 69 75 77 10 17 22 27 40 45 46 53 $\textbf{V}_{12}$ 12 14 30 32 48 50 75 77 20 22 24 27 38 40 42 45 $\textbf{V}_{13}$ 47 50 56 59 65 68 74 77 15 16 17 18 42 43 44 45 $\textbf{V}_{14}$ 25 26 34 35 47 50 74 77 15 18 42 45 57 58 66 67 $\textbf{V}_{15}$ 19 23 28 32 64 68 73 77 3 4 8 9 39 40 44 45 $\textbf{V}_{16}$ 19 23 49 53 58 62 73 77 3 9 33 34 39 45 69 70
 1 -1 vector coordinate $\textbf{V}_1$ 2 3 7 8 74 75 79 80 10 18 19 27 55 63 64 72 $\textbf{V}_2$ 10 12 16 18 64 66 70 72 2 8 20 26 56 62 74 80 $\textbf{V}_3$ 28 29 35 36 46 47 53 54 4 6 13 15 67 69 76 78 $\textbf{V}_4$ 4 8 24 25 40 44 78 79 28 32 48 54 57 63 64 68 $\textbf{V}_5$ 37 40 47 54 55 58 65 72 5 7 15 17 32 34 78 80 $\textbf{V}_6$ 4 9 12 14 48 50 67 72 20 24 28 35 38 42 73 80 $\textbf{V}_7$ 19 27 29 34 38 43 64 72 3 8 13 14 58 59 75 80 $\textbf{V}_8$ 37 39 43 45 46 48 52 54 5 6 23 24 59 60 77 78 $\textbf{V}_9$ 2 8 20 26 43 45 52 54 10 12 59 60 64 66 77 78 $\textbf{V}_{10}$ 47 48 49 50 74 75 76 77 15 18 24 27 33 36 42 45 $\textbf{V}_{11}$ 2 6 30 32 65 69 75 77 10 17 22 27 40 45 46 53 $\textbf{V}_{12}$ 12 14 30 32 48 50 75 77 20 22 24 27 38 40 42 45 $\textbf{V}_{13}$ 47 50 56 59 65 68 74 77 15 16 17 18 42 43 44 45 $\textbf{V}_{14}$ 25 26 34 35 47 50 74 77 15 18 42 45 57 58 66 67 $\textbf{V}_{15}$ 19 23 28 32 64 68 73 77 3 4 8 9 39 40 44 45 $\textbf{V}_{16}$ 19 23 49 53 58 62 73 77 3 9 33 34 39 45 69 70
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