Recursive parametrization and invariant phases of unitary matrices

Some Formulas for Invariant Phases of Unitary Matrices by Jarlskog

CP violation is expected in the standard model of particle physics with three or more families [1], [2]. Therefore it is an important problem that what is the measure of CP violation with such families which is invariant under the action of phase factors. To construct invariants for matrix action, the determinant is a useful tool. In the previous paper [3], C. Jarlskog succeeded to define invar...

Rephasing invariant parametrization of flavor mixing matrices

Jarlskog’s Parametrization of Unitary Matrices and Qudit Theory

In the paper (math–ph/0504049) Jarlskog gave an interesting simple parametrization to unitary matrices, which was essentially the canonical coordinate of the second kind in the Lie group theory (math–ph/0505047). In this paper we apply the method to a quantum computation based on multi– level system (qudit theory). Namely, by considering that the parametrization gives a complete set of modules ...

Unitary Matrices and Hermitian Matrices

Unitary Matrices and Hermitian Matrices Recall that the conjugate of a complex number a + bi is a − bi. The conjugate of a + bi is denoted a+ bi or (a+ bi)∗. In this section, I’ll use ( ) for complex conjugation of numbers of matrices. I want to use ( )∗ to denote an operation on matrices, the conjugate transpose. Thus, 3 + 4i = 3− 4i, 5− 6i = 5 + 6i, 7i = −7i, 10 = 10. Complex conjugation sati...

Graphs of unitary matrices

The support of a matrix M is the (0, 1)-matrix with ij-th entry equal to 1 if the ij-th entry of M is non-zero, and equal to 0, otherwise. The digraph whose adjacency matrix is the support of M is said to be the digraph of M . This paper observes some structural properties of digraphs and Cayley digraphs, of unitary matrices. We prove that a group generated by two elements has a set of generato...