We give a one-to-one correspondence between classes of density matrices under local unitary invariance and the double cosets of unitary groups. We show that the interrelationship among classes of local unitary equivalent multi-partite mixed states is independent from the actual values of the eigenvalues and only depends on the multiplicities of the eigenvalues. The interpretation in terms of ho...

This paper presents a Σ-unitary analogue to the CS decomposition of a partitioned unitary matrix. The hyperbolic rotations revealed by the decomposition are shown to be optimal in that, among a broader class of decompositions of Σ-unitary matrices into elementary hyperbolic rotations, they are the smallest possible in a sum-of-squares sense. A similar optimality property is shown to hold for th...

We classify local unitary equivalence classes of symmetric states via a classification of their local unitary stabilizer subgroups. For states whose local unitary stabilizer groups have a positive number of continuous degrees of freedom, the classification is exhaustive. We show that local unitary stabilizer groups with no continuous degrees of freedom are isomorphic to finite subgroups of the ...

Quantum circuit is generalized to include non-unitary gates which are operated by quantum measurement. A 1-qubit non-unitary gate, together with the controlled-NOT gate and the 1-qubit unitary gate, is shown to constitute a universal set of gates for the non-unitary quantum circuit that reduces the number of the overhead of qubits required to ensure fault tolerance. A reversing measurement sche...

Let  be a commutative ring and  be a unitary  module. We define a semi prime sub module of a module and consider various properties of it. Also we define semi-radical of a sub module of a module and give a number of its properties. We define modules which satisfy the semi-radical formula and present the existence of such a module.

Some aspects of the perturbation theory for eigenvalues of unitary matrices are considered. Making use of the close relation between unitary and Hermitian eigenvalue problems a Courant-Fischer-type theorem for unitary matrices is derived and an inclusion theorem analogue to the Kahan theorem for Hermitian matrices is presented. Implications for the special case of unitary Hessenberg matrices ar...

Circular Brownian motion models of random matrices were introduced by Dyson and describe the parametric eigenparameter correlations of unitary random matrices. For symmetric unitary, self-dual quaternion unitary and an analogue of antisymmetric hermitian matrix initial conditions, Brownian dynamics toward the unitary symmetry is analyzed. The dynamical correlation functions of arbitrary number ...

Motivated by easy access to complete corepresentation (corep) data of all the 1651 magnetic space groups (MSGs) in three-dimensional space, we have developed a Mathematica package MSGCorep provide an offline database coreps and various functions manipulate them, based on our previous SpaceGroupIrep. One can use obtain elements any MSG little group, calculate multiplication group elements, small...

We consider the partition function for a matrix model with a global unitary invariant energy function. We show that the averages over the partition function of global unitary invariant trace polynomials of the matrix variables are the same when calculated with any choice of a global unitary fixing, while averages of such polynomials without a trace define matrixvalued correlation functions, tha...

Two possible quantum descriptions of the classical Not gate are investigated in the framework of the Hilbert space C: the unitary and the anti–unitary operator realizations. The two cases are distinguished interpreting the unitary Not as a quantum realization of the classical gate which on a fixed orthogonal pair of unit vectors, realizing once for all the classical bits 0 and 1, produces the r...