20,681 research outputs found
The Convex Closure of the Output Entropy of Infinite Dimensional Channels and the Additivity Problem
The continuity properties of the convex closure of the output entropy of
infinite dimensional channels and their applications to the additivity problem
are considered.
The main result of this paper is the statement that the superadditivity of
the convex closure of the output entropy for all finite dimensional channels
implies the superadditivity of the convex closure of the output entropy for all
infinite dimensional channels, which provides the analogous statements for the
strong superadditivity of the EoF and for the additivity of the minimal output
entropy.
The above result also provides infinite dimensional generalization of Shor's
theorem stated equivalence of different additivity properties.
The superadditivity of the convex closure of the output entropy (and hence
the additivity of the minimal output entropy) for two infinite dimensional
channels with one of them a direct sum of noiseless and entanglement-breaking
channels are derived from the corresponding finite dimensional results.
In the context of the additivity problem some observations concerning
complementary infinite dimensional channels are considered.Comment: 24 page
Representation of non-semibounded quadratic forms and orthogonal additivity
In this article we give a representation theorem for non-semibounded
Hermitean quadratic forms in terms of a (non-semibounded) self-adjoint
operator. The main assumptions are closability of the Hermitean quadratic form,
the direct integral structure of the underlying Hilbert space and orthogonal
additivity. We apply this result to several examples, including the position
operator in quantum mechanics and quadratic forms invariant under a unitary
representation of a separable locally compact group. The case of invariance
under a compact group is also discussed in detail
Superadditivity in trade-off capacities of quantum channels
In this article, we investigate the additivity phenomenon in the dynamic
capacity of a quantum channel for trading classical communication, quantum
communication and entanglement. Understanding such additivity property is
important if we want to optimally use a quantum channel for general
communication purpose. However, in a lot of cases, the channel one will be
using only has an additive single or double resource capacity, and it is
largely unknown if this could lead to an superadditive double or triple
resource capacity. For example, if a channel has an additive classical and
quantum capacity, can the classical-quantum capacity be superadditive? In this
work, we answer such questions affirmatively.
We give proof-of-principle requirements for these channels to exist. In most
cases, we can provide an explicit construction of these quantum channels. The
existence of these superadditive phenomena is surprising in contrast to the
result that the additivity of both classical-entanglement and classical-quantum
capacity regions imply the additivity of the triple capacity region.Comment: 15 pages. v2: typo correcte
Stability of Kronecker coefficients via discrete tomography
In this paper we give a new sufficient condition for a general stability of
Kronecker coefficients, which we call it additive stability. It was motivated
by a recent talk of J. Stembridge at the conference in honor of Richard P.
Stanley's 70th birthday, and it is based on work of the author on discrete
tomography along the years. The main contribution of this paper is the
discovery of the connection between additivity of integer matrices and
stability of Kronecker coefficients. Additivity, in our context, is a concept
from discrete tomography. Its advantage is that it is very easy to produce lots
of examples of additive matrices and therefore of new instances of stability
properties. We also show that Stembridge's hypothesis and additivity are
closely related, and prove that all stability properties of Kronecker
coefficients discovered before fit into additive stability.Comment: 22 page
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