79 research outputs found
Determinantal and eigenvalue inequalities for matrices with numerical ranges in a sector
Let A = \pmatrix A_{11} & A_{12} \cr A_{21} & A_{22}\cr\pmatrix \in M_n,
where with , be such that the numerical range of
lies in the set \{e^{i\varphi} z \in \IC: |\Im z| \le (\Re z) \tan
\alpha\}, for some and . We
obtain the optimal containment region for the generalized eigenvalue
satisfying \lambda \pmatrix A_{11} & 0 \cr 0 & A_{22}\cr\pmatrix x = \pmatrix
0 & A_{12} \cr A_{21} & 0\cr\pmatrix x \quad \hbox{for some nonzero} x \in
\IC^n, and the optimal eigenvalue containment region of the matrix in case and are
invertible. From this result, one can show . In particular, if is a accretive-dissipative
matrix, then . These affirm some
conjectures of Drury and Lin.Comment: 6 pages, to appear in Journal of Mathematical Analysi
Canonical forms, higher rank numerical range, convexity, totally isotropic subspace, matrix equations
Results on matrix canonical forms are used to give a complete description of
the higher rank numerical range of matrices arising from the study of quantum
error correction. It is shown that the set can be obtained as the intersection
of closed half planes (of complex numbers). As a result, it is always a convex
set in . Moreover, the higher rank numerical range of a normal
matrix is a convex polygon determined by the eigenvalues. These two
consequences confirm the conjectures of Choi et al. on the subject. In
addition, the results are used to derive a formula for the optimal upper bound
for the dimension of a totally isotropic subspace of a square matrix, and
verify the solvability of certain matrix equations.Comment: 10 pages. To appear in Proceedings of the American Mathematical
Societ
Linear rank preservers of tensor products of rank one matrices
Let be integers larger than or equal to 2. We characterize
linear maps such that
Applying
this result, we extend two recent results on linear maps that preserving the
rank of special classes of matrices.Comment: 12 page
A note on the perturbation of positive matrices by normal and unitary matrices
AbstractIn a recent paper, Neumann and Sze considered for an n×n nonnegative matrix A, the minimization and maximization of ρ(A+S), the spectral radius of (A+S), as S ranges over all the doubly stochastic matrices. They showed that both extremal values are always attained at an n×n permutation matrix. As a permutation matrix is a particular case of a normal matrix whose spectral radius is 1, we consider here, for positive matrices A such that (A+N) is a nonnegative matrix, for all normal matrices N whose spectral radius is 1, the minimization and maximization problems of ρ(A+N) as N ranges over all such matrices. We show that the extremal values always occur at an n×n real unitary matrix. We compare our results with a less recent work of Han, Neumann, and Tastsomeros in which the maximum value of ρ(A+X) over all n×n real matrices X of Frobenius norm n was sought
Linear preservers and quantum information science
Let be positive integers, the set of complex
matrices and the set of complex matrices. Regard as
the tensor space . Suppose is the Ky Fan -norm
with , or the Schatten -norm with
() on . It is shown that a linear map satisfying for all
and if and only if there are unitary such that
has the form ,
where is either the identity map or the
transposition map . The results are extended to tensor space
of higher level. The connection of the
problem to quantum information science is mentioned.Comment: 13 page
Characterizations of inverse M-matrices with special zero patterns
AbstractIn this paper, we provide some characterizations of inverse M-matrices with special zero patterns. In particular, we give necessary and sufficient conditions for k-diagonal matrices and symmetric k-diagonal matrices to be inverse M-matrices. In addition, results for triadic matrices, tridiagonal matrices and symmetric 5-diagonal matrices are presented as corollaries
Maximal noiseless code rates for collective rotation channels on qudits
We study noiseless subsystems on collective rotation channels of qudits,
i.e., quantum channels with operators in the set This is done by analyzing the
decomposition of the algebra generated by . We summarize the results for the channels on qubits (), and
obtain the maximum dimension of the noiseless subsystem that can be used as the
quantum error correction code for the channel. Then we extend our results to
general . In particular, it is shown that the code rate, i.e., the number of
protected qudits over the number of physical qudits, always approaches 1 for a
suitable noiseless subsystem. Moreover, one can determine the maximum dimension
of the noiseless subsystem by solving a non-trivial discrete optimization
problem. The maximum dimension of the noiseless subsystem for (qutrits)
is explicitly determined by a combination of mathematical analysis and the
symbolic software Mathematica.Comment: 16 pages, proofs are put in Appendix for clearer presentation. Title
has been changed and some related materials, such as quantum secret sharing
and erasure errors, are mentione
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