The aim of this paper is to study linear preservers of the trace of Kronecker sums and their connection with preservers of determinants of Kronecker products. The partial trace and partial determinant play a fundamental role in characterizing the pre
servers of the trace of Kronecker sums and preservers of the determinant of Kronecker products respectively.
Let $det_2(A)$ be the block-wise determinant (partial determinant). We consider the condition for completing the determinant $det(det_2(A)) = det(A),$ and characterize the case for an arbitrary Kronecker product $A$ of matrices over an arbitrary fiel
d. Further insisting that $det_2(AB)=det_2(A)det_2(B)$, for Kronecker products $A$ and $B$, yields a multiplicative monoid of matrices. This leads to a determinant-root operation $text{Det}$ which satisfies $text{Det}(text{Det}_2(A)) = text{Det}(A)$ when $A$ is a Kronecker product of matrices for which $text{Det}$ is defined.
We consider the vector space of $n times n$ matrices over $mathbb C$, Fermi operators and operators constructed from these matrices and Fermi operators. The properties of these operators are studied with respect to the underlying matrices. The commut
ators, anticommutators, and the eigenvalue problem of such operators are also discussed. Other matrix functions such as the exponential functions are studied. Density operators and Kraus operators are also discussed.
We study truncated Bose operators in finite dimensional Hilbert spaces. Spin coherent states for the truncated Bose operators and canonical coherent states for Bose operators are compared. The Lie algebra structure and the spectrum of the truncated Bose operators are discussed.
{bf Abstract.} We show that two hierarchies of spin Hamilton operators admit the same spectrum. Both Hamilton operators play a central role for quantum gates in particular for the case spin-$frac12$ and the case spin-1. The spin-$frac12$, spin-1, spi
n-$frac32$ and spin-2 cases are studied in detail. Entanglement and mutually unbiased bases of the eigenvectors is discussed. Two triple Hamilton operators are also investigated. Both are also admitting the same spectrum.
The results of [I. Ojeda, Amer. Math. Monthly, 122, pp 60--64] provides a characterization of Kronecker square roots of matrices in terms of the symmetry and rank of the block vec matrix (rearrangement matrix). In this short note we reformulate the c
haracterization in terms of rank only by considering an alternative to the block vec matrix, provided that the characteristic of the underlying field is not equal to 2.
Pauli spin matrices, Pauli group, commutators, anti-commutators and the Kronecker product are studied. Applications to eigenvalue problems, exponential functions of such matrices, spin Hamilton operators, mutually unbiased bases, Fermi operators and Bose operators are provided.
Leopardi introduced the notion of a Kronecker quotient in [Paul Leopardi. A generalized FFT for Clifford algebras. Bulletin of the Belgian Mathematical Society, 11:663--688, 2005.]. This article considers the basic properties that a Kronecker quotien
t should satisfy and additional properties which may be satisfied. A class of Kronecker quotients for which these properties have a natural description is completely characterized. Two examples of types of Kronecker quotients are described.
We describe the construction of quantum gates (unitary operators) from boolean functions and give a number of applications. Both non-reversible and reversible boolean functions are considered. The construction of the Hamilton operator for a quantum g
ate is also described with the Hamilton operator expressed as spin system. Computer algebra implementations are provided.
We describe solutions of the matrix equation $exp(z(A-I_n))=A$, where $z in {mathbb C}$. Applications in quantum computing are given. Both normal and nonnormal matrices are studied. For normal matrices, the Lambert W-function plays a central role.
