280 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 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
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
Physical transformations between quantum states
Given two sets of quantum states {A_1, ..., A_k} and {B_1, ..., B_k},
represented as sets of density matrices, necessary and sufficient conditions
are obtained for the existence of a physical transformation T, represented as a
trace-preserving completely positive map, such that T(A_i) = B_i for i = 1,
..., k. General completely positive maps without the trace-preserving
requirement, and unital completely positive maps transforming the states are
also considered
Recursive encoding and decoding of the noiseless subsystem for qudits
We give a full explanation of the noiseless subsystem that protects a
single-qubit against collective errors and the corresponding recursive scheme
described by C.-K. Li et. al. [Phys. Rev. A 84, 044301 (2011)] from a
representation theory point of view. Furthermore, we extend the construction to
qudits under the influence of collective SU() errors. We find that under
this recursive scheme, the asymptotic encoding rate is
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