57 research outputs found
Symplectic analogs of polar decomposition and their applications to bosonic Gaussian channels
We obtain several analogs of real polar decomposition for even dimensional
matrices. In particular, we decompose a non-degenerate matrix as a product of a
Hamiltonian and an anti-symplectic matrix and under additional requirements we
decompose a matrix as a skew-Hamiltonian and a symplectic matrix. We apply our
results to study bosonic Gaussian channels up to inhomogeneous symplectic
transforms
The Pad\'e iterations for the matrix sign function and their reciprocals are optimal
It is proved that among the rational iterations locally converging with order
s>1 to the sign function, the Pad\'e iterations and their reciprocals are the
unique rationals with the lowest sum of the degrees of numerator and
denominator
Scalable Task-Based Algorithm for Multiplication of Block-Rank-Sparse Matrices
A task-based formulation of Scalable Universal Matrix Multiplication
Algorithm (SUMMA), a popular algorithm for matrix multiplication (MM), is
applied to the multiplication of hierarchy-free, rank-structured matrices that
appear in the domain of quantum chemistry (QC). The novel features of our
formulation are: (1) concurrent scheduling of multiple SUMMA iterations, and
(2) fine-grained task-based composition. These features make it tolerant of the
load imbalance due to the irregular matrix structure and eliminate all
artifactual sources of global synchronization.Scalability of iterative
computation of square-root inverse of block-rank-sparse QC matrices is
demonstrated; for full-rank (dense) matrices the performance of our SUMMA
formulation usually exceeds that of the state-of-the-art dense MM
implementations (ScaLAPACK and Cyclops Tensor Framework).Comment: 8 pages, 6 figures, accepted to IA3 2015. arXiv admin note: text
overlap with arXiv:1504.0504
On computing complex square roots of real matrices
We present an idea for computing complex square roots of matrices using only real
arithmetic.Fundação para a Ciência e a Tecnologia (FCT) - Project PEst-C/MAT/UI0013/2011, PTDC/MAT/112273/2009National Natural Science Foundation of China - Project PEst-C/MAT/UI0013/2011, PTDC/MAT/112273/2009, Portugal.Major Foundation of Educational Committee of Hunan Province - Grant No. 09A002 [2009]FEDER Funds- "Programa Operacional Factores de Competitividade - COMPETE
New method for square root of non-singular M-matrix
Square root of a matrix play an important role in many applications of matrix theory. In this paper, we propose a new iterative method for square root of a non-singular M-matrix. We first transform the matrix equation X2 – A=0 into special form of a non-symmetric algebraic Riccati equation (NARE), and then solve this special NARE by Newton method. Efficiency and effectiveness proved by theoretical analysis and numerical experiments. Keywords: - Matrix square root, M-matrix, Non-symmetric algebraic Riccati equation, Newton method
Classical and nonclassical randomness in quantum measurements
The space of positive operator-valued measures on the Borel sets of a compact
(or even locally compact) Hausdorff space with values in the algebra of linear
operators acting on a d-dimensional Hilbert space is studied from the
perspectives of classical and non-classical convexity through a transform
that associates any positive operator-valued measure with a certain
completely positive linear map of the homogeneous C*-algebra
into . This association is achieved by using an operator-valued integral
in which non-classical random variables (that is, operator-valued functions)
are integrated with respect to positive operator-valued measures and which has
the feature that the integral of a random quantum effect is itself a quantum
effect. A left inverse for yields an integral representation,
along the lines of the classical Riesz Representation Theorem for certain
linear functionals on , of certain (but not all) unital completely
positive linear maps . The extremal and
C*-extremal points of the space of POVMS are determined.Comment: to appear in Journal of Mathematical Physic
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