A characterization of orthogonal permutative matrices of order 4 (Q2080247)

This paper contributes to a complete parametric characterization of all complex, real, and rational Orthogonal Permutative Matrices (OPMs) of order 4, where the OPMs can be expressed as a linear combination of up to 4 permutation matrices. Furthermore, the authors construct the matrix spaces generated by such permutation matrices in a way that orthogonal matrices in those spaces are always permutative or expressed as the direct sum of OPMs up to the permutation of rows and columns of the matrices. From the point of view of quantum Shannon theory, a very interesting aspect is the characterization and classification of OPMs and Grover's diffusion operators. The findings of this great paper offer a typical example of basic scientific knowledge which can be used in applied sciences, as quantum engineering in this concrete case. In this context, a fundamental question related to the ``quantum noisy channel coding theorem'' is the following: is it possible to resolve the quantum channel capacity problem with the authors' method? This paper shows that a key for a breakthrough, as it often happens, is the cooperation among different branches of sciences. Furthermore the obtained results let us hope that there is a chance for a generalization in the case of unitary groups, which in turn may be used to study unsolved problems in quantum Shannon theory.