7,141 research outputs found
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
Entropic trade-off relations for quantum operations
Spectral properties of an arbitrary matrix can be characterized by the
entropy of its rescaled singular values. Any quantum operation can be described
by the associated dynamical matrix or by the corresponding superoperator. The
entropy of the dynamical matrix describes the degree of decoherence introduced
by the map, while the entropy of the superoperator characterizes the a priori
knowledge of the receiver of the outcome of a quantum channel Phi. We prove
that for any map acting on a N--dimensional quantum system the sum of both
entropies is not smaller than ln N. For any bistochastic map this lower bound
reads 2 ln N. We investigate also the corresponding R\'enyi entropies,
providing an upper bound for their sum and analyze entanglement of the
bi-partite quantum state associated with the channel.Comment: 10 pages, 4 figure
An Overview of Polynomially Computable Characteristics of Special Interval Matrices
It is well known that many problems in interval computation are intractable,
which restricts our attempts to solve large problems in reasonable time. This
does not mean, however, that all problems are computationally hard. Identifying
polynomially solvable classes thus belongs to important current trends. The
purpose of this paper is to review some of such classes. In particular, we
focus on several special interval matrices and investigate their convenient
properties. We consider tridiagonal matrices, {M,H,P,B}-matrices, inverse
M-matrices, inverse nonnegative matrices, nonnegative matrices, totally
positive matrices and some others. We focus in particular on computing the
range of the determinant, eigenvalues, singular values, and selected norms.
Whenever possible, we state also formulae for determining the inverse matrix
and the hull of the solution set of an interval system of linear equations. We
survey not only the known facts, but we present some new views as well
Split Quaternions and Particles in (2+1)-Space
It is known that quaternions represent rotations in 3D Euclidean and
Minkowski spaces. However, product by a quaternion gives rotation in two
independent planes at once and to obtain single-plane rotations one has to
apply by half-angle quaternions twice from the left and on the right (with its
inverse). This 'double cover' property is potential problem in geometrical
application of split quaternions, since (2+2)-signature of their norms should
not be changed for each product. If split quaternions form proper algebraic
structure for microphysics, representation of boosts in (2+1)-space leads to
the interpretation of the scalar part of quaternions as wavelength of
particles. Invariance of space-time intervals and some quantum behavior, like
noncommutativity and fundamental spinor representation, probably also are
algebraic properties. In our approach the Dirac equation represents the
Cauchy-Riemann analyticity condition and the two fundamental physical
parameters (speed of light and Planck's constant) appear from the requirement
of positive definiteness of quaternionic norms.Comment: The version published in Eur. Phys. J.
Tsirelson's problem and Kirchberg's conjecture
Tsirelson's problem asks whether the set of nonlocal quantum correlations
with a tensor product structure for the Hilbert space coincides with the one
where only commutativity between observables located at different sites is
assumed. Here it is shown that Kirchberg's QWEP conjecture on tensor products
of C*-algebras would imply a positive answer to this question for all bipartite
scenarios. This remains true also if one considers not only spatial
correlations, but also spatiotemporal correlations, where each party is allowed
to apply their measurements in temporal succession; we provide an example of a
state together with observables such that ordinary spatial correlations are
local, while the spatiotemporal correlations reveal nonlocality. Moreover, we
find an extended version of Tsirelson's problem which, for each nontrivial Bell
scenario, is equivalent to the QWEP conjecture. This extended version can be
conveniently formulated in terms of steering the system of a third party.
Finally, a comprehensive mathematical appendix offers background material on
complete positivity, tensor products of C*-algebras, group C*-algebras, and
some simple reformulations of the QWEP conjecture.Comment: 57 pages, to appear in Rev. Math. Phy
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