B. S. Tsirelson showed that comparisons between probabilities in “classical” physics and probabilities in quantum mechanics yield discrepancy measures $K_n $ for finite $n \times n$ real matrices that approach Grothendieck’s constant $K_G $ as n gets large. It is known that $K_2 = K_3 = \sqrt{2} $ and that $K_G \geq \pi /2 = 1.57 \cdots $, but examples of $n \times n$ matrices for specified n that demonstrate $K_n > \sqrt{2} $ have eluded researchers. A series of elementary examples are provided, which yield lower bounds on $K_{k ( k - 1 )} $ that approach 3/2 as k gets large. A uniform change along the main diagonal of our basic example shows that $K_{20} \geqq = 10/7 = 1.42 \cdots $.B. S. Tsirelson showed that comparisons between probabilities in “classical” physics and probabilities in quantum mechanics yield discrepancy measures $K_n $ for finite $n \times n$ real matrices that approach Grothendieck’s constant $K_G $ as n gets large. It is known that $K_2 = K_3 = \sqrt{2} $ and that $K_G \geq \pi /2 = 1.57 \cdots $, but examples of $n \times n$ matrices for specified n that demonstrate $K_n > \sqrt{2} $ have eluded researchers. A series of elementary examples are provided, which yield lower bounds on $K_{k ( k - 1 )} $ that approach 3/2 as k gets large. A uniform change along the main diagonal of our basic example shows that $K_{20} \geqq = 10/7 = 1.42 \cdots $.

SIAM Journal on Discrete MathematicsFeb 1, 199450 citations

Eigenvalues and eigenvectors of complex Hadamard matrices

Mengfan Liang, Lin Chen

Characterizing the $6\times 6$ complex Hadamard matrices (CHMs) is an open problem in linear algebra and quantum information. In this paper, we investigate the eigenvalues and eigenvectors of CHMs. We show that any $n\times n$ CHM with dephased form has two constant eigenvalues $\pm\sqrt{n}$ and has two constant eigenvectors. We obtain the maximum numbers of identical eigenvalues of $6\times 6$ CHMs with dephased form and we extend this result to arbitrary dimension. We also show that there is no $6\times 6$ CHM with four identical eigenvalues. We conjecture that the eigenvalues and eigenvectors of $6\times 6$ CHMs will lead to the complete classification of $6\times 6$ CHMs.Characterizing the $6\times 6$ complex Hadamard matrices (CHMs) is an open problem in linear algebra and quantum information. In this paper, we investigate the eigenvalues and eigenvectors of CHMs. We show that any $n\times n$ CHM with dephased form has two constant eigenvalues $\pm\sqrt{n}$ and has two constant eigenvectors. We obtain the maximum numbers of identical eigenvalues of $6\times 6$ CHMs with dephased form and we extend this result to arbitrary dimension. We also show that there is no $6\times 6$ CHM with four identical eigenvalues. We conjecture that the eigenvalues and eigenvectors of $6\times 6$ CHMs will lead to the complete classification of $6\times 6$ CHMs.

arXiv:2408.10471Aug 20, 20240 citations

p-adic dynamical systems

P. Freund, Mark Olson

Nuclear Physics BFeb 1, 198846 citations

Mahler/Zeta Correspondence

Takashi Komatsu, Norio Konno, Iwao Sato +1 more

arXiv:2202.05966Quantum Information ProcessingAug 20, 20223 citations

Unbiased Hadamard matrices and bases

Hadi Kharaghani

Mutually unbiased unit Hadamard (MUUH) matrices have been studied for almost 40 years. Recent interest in such matrices has been motivated by their applications to quantum information theory. In this paper, we introduce the class of mutually unbiased complex Hadamard (MUCH) matrices having as sntries fourth roots of unity. The number of MUCH matrices of order 2n, n odd, is at most 2, and the bound is attained for n = 1,5,9. Certain pairs of mutually unbiased complex Hadamard matrices of order m can be used to construct pairs of unbiased real Hadamard matrices of order 2m. We then turn our attention to the class of mutually unbiased real Hadamard matrices of order 16n2which seem to be the most interesting ones. Mutually unbiased weighing (MUW) matrices are introduced for the first time. Further study of these objects look quite promising.Mutually unbiased unit Hadamard (MUUH) matrices have been studied for almost 40 years. Recent interest in such matrices has been motivated by their applications to quantum information theory. In this paper, we introduce the class of mutually unbiased complex Hadamard (MUCH) matrices having as sntries fourth roots of unity. The number of MUCH matrices of order 2n, n odd, is at most 2, and the bound is attained for n = 1,5,9. Certain pairs of mutually unbiased complex Hadamard matrices of order m can be used to construct pairs of unbiased real Hadamard matrices of order 2m. We then turn our attention to the class of mutually unbiased real Hadamard matrices of order 16n2which seem to be the most interesting ones. Mutually unbiased weighing (MUW) matrices are introduced for the first time. Further study of these objects look quite promising.

NATO science for peace and security series. D, Information and communication securityJan 1, 20111 citation

B. S. Tsirelson showed that comparisons between probabilities in “classical” physics and probabilities in quantum mechanics yield discrepancy measures $K_n $ for finite

n \times n

real matrices that approach Grothendieck’s constant $K_G $ as n gets large. It is known that $K_2 = K_3 = \sqrt{2} $ and that $K_G \geq \pi /2 = 1.57 \cdots $, but examples of

n \times n

matrices for specified n that demonstrate $K_n > \sqrt{2} $ have eluded researchers. A series of elementary examples are provided, which yield lower bounds on $K_{k ( k - 1 )} $ that approach 3/2 as k gets large. A uniform change along the main diagonal of our basic example shows that $K_{20} \geqq = 10/7 = 1.42 \cdots $.

Characterizing the

6\times 6

complex Hadamard matrices (CHMs) is an open problem in linear algebra and quantum information. In this paper, we investigate the eigenvalues and eigenvectors of CHMs. We show that any

n\times n

CHM with dephased form has two constant eigenvalues

\pm\sqrt{n}

and has two constant eigenvectors. We obtain the maximum numbers of identical eigenvalues of

6\times 6

CHMs with dephased form and we extend this result to arbitrary dimension. We also show that there is no

6\times 6

CHM with four identical eigenvalues. We conjecture that the eigenvalues and eigenvectors of

6\times 6

CHMs will lead to the complete classification of

6\times 6

CHMs.