# 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

show all references

##### 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
 [1] Tao Wu, Yu Lei, Jiao Shi, Maoguo Gong. An evolutionary multiobjective method for low-rank and sparse matrix decomposition. Big Data & Information Analytics, 2017, 2 (1) : 23-37. doi: 10.3934/bdia.2017006 [2] Charles Fulton, David Pearson, Steven Pruess. Characterization of the spectral density function for a one-sided tridiagonal Jacobi matrix operator. Conference Publications, 2013, 2013 (special) : 247-257. doi: 10.3934/proc.2013.2013.247 [3] Yuri Fedorov, Božidar Jovanović. Continuous and discrete Neumann systems on Stiefel varieties as matrix generalizations of the Jacobi–Mumford systems. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2559-2599. doi: 10.3934/dcds.2020375 [4] Takeshi Saito, Kazuyuki Yagasaki. Chebyshev spectral methods for computing center manifolds. Journal of Computational Dynamics, 2021  doi: 10.3934/jcd.2021008 [5] M. Phani Sudheer, Ravi S. Nanjundiah, A. S. Vasudeva Murthy. Revisiting the slow manifold of the Lorenz-Krishnamurthy quintet. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1403-1416. doi: 10.3934/dcdsb.2006.6.1403 [6] Hua Shi, Xiang Zhang, Yuyan Zhang. Complex planar Hamiltonian systems: Linearization and dynamics. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3295-3317. doi: 10.3934/dcds.2020406 [7] Weisong Dong, Chang Li. Second order estimates for complex Hessian equations on Hermitian manifolds. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2619-2633. doi: 10.3934/dcds.2020377 [8] Wei Xi Li, Chao Jiang Xu. Subellipticity of some complex vector fields related to the Witten Laplacian. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021047 [9] Simão Correia, Mário Figueira. A generalized complex Ginzburg-Landau equation: Global existence and stability results. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021056 [10] Joel Coacalle, Andrew Raich. Compactness of the complex Green operator on non-pseudoconvex CR manifolds. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021061 [11] Yohei Yamazaki. Center stable manifolds around line solitary waves of the Zakharov–Kuznetsov equation with critical speed. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3579-3614. doi: 10.3934/dcds.2021008 [12] Huy Dinh, Harbir Antil, Yanlai Chen, Elena Cherkaev, Akil Narayan. Model reduction for fractional elliptic problems using Kato's formula. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021004 [13] Xiaofei Liu, Yong Wang. Weakening convergence conditions of a potential reduction method for tensor complementarity problems. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021080 [14] Palash Sarkar, Subhadip Singha. Classical reduction of gap SVP to LWE: A concrete security analysis. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021004 [15] Marian Gidea, Rafael de la Llave, Tere M. Seara. A general mechanism of instability in Hamiltonian systems: Skipping along a normally hyperbolic invariant manifold. Discrete & Continuous Dynamical Systems, 2020, 40 (12) : 6795-6813. doi: 10.3934/dcds.2020166 [16] Dingheng Pi. Periodic orbits for double regularization of piecewise smooth systems with a switching manifold of codimension two. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021080 [17] Changpin Li, Zhiqiang Li. Asymptotic behaviors of solution to partial differential equation with Caputo–Hadamard derivative and fractional Laplacian: Hyperbolic case. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021023 [18] Peter Benner, Jens Saak, M. Monir Uddin. Balancing based model reduction for structured index-2 unstable descriptor systems with application to flow control. Numerical Algebra, Control & Optimization, 2016, 6 (1) : 1-20. doi: 10.3934/naco.2016.6.1 [19] Guiyang Zhu. Optimal pricing and ordering policy for defective items under temporary price reduction with inspection errors and price sensitive demand. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021060 [20] Ruonan Liu, Run Xu. Hermite-Hadamard type inequalities for harmonical $(h1,h2)-$convex interval-valued functions. Mathematical Foundations of Computing, 2021  doi: 10.3934/mfc.2021005

2019 Impact Factor: 1.233