11 research outputs found
On Shor's channel extension and constrained channels
In this paper we give several equivalent formulations of the additivity
conjecture for constrained channels, which formally is substantially stronger
than the unconstrained additivity. To this end a characteristic property of the
optimal ensemble for such a channel is derived, generalizing the maximal
distance property. It is shown that the additivity conjecture for constrained
channels holds true for certain nontrivial classes of channels.
Recently P. Shor showed that conjectured additivity properties for several
quantum information quantities are in fact equivalent. After giving an
algebraic formulation for the Shor's channel extension, its main asymptotic
property is proved. It is then used to show that additivity for two constrained
channels can be reduced to the same problem for unconstrained channels, and
hence, "global" additivity for channels with arbitrary constraints is
equivalent to additivity without constraints.Comment: 19 pages; substantially revised and enhanced. To appear in Commun.
Math. Phy
Continuity of the von Neumann entropy
A general method for proving continuity of the von Neumann entropy on subsets
of positive trace-class operators is considered. This makes it possible to
re-derive the known conditions for continuity of the entropy in more general
forms and to obtain several new conditions. The method is based on a particular
approximation of the von Neumann entropy by an increasing sequence of concave
continuous unitary invariant functions defined using decompositions into finite
rank operators. The existence of this approximation is a corollary of a general
property of the set of quantum states as a convex topological space called the
strong stability property. This is considered in the first part of the paper.Comment: 42 pages, the minor changes have been made, the new applications of
the continuity condition have been added. To appear in Commun. Math. Phy
Information capacity of quantum observable
In this paper we consider the classical capacities of quantum-classical
channels corresponding to measurement of observables. Special attention is paid
to the case of continuous observables. We give the formulas for unassisted and
entanglement-assisted classical capacities and consider some
explicitly solvable cases which give simple examples of entanglement-breaking
channels with We also elaborate on the ensemble-observable duality
to show that for the measurement channel is related to the
-quantity for the dual ensemble in the same way as is related to the
accessible information. This provides both accessible information and the
-quantity for the quantum ensembles dual to our examples.Comment: 13 pages. New section and references are added concerning the
ensemble-observable dualit
Continuity of quantum channel capacities
We prove that a broad array of capacities of a quantum channel are
continuous. That is, two channels that are close with respect to the diamond
norm have correspondingly similar communication capabilities. We first show
that the classical capacity, quantum capacity, and private classical capacity
are continuous, with the variation on arguments epsilon apart bounded by a
simple function of epsilon and the channel's output dimension. Our main tool is
an upper bound of the variation of output entropies of many copies of two
nearby channels given the same initial state; the bound is linear in the number
of copies. Our second proof is concerned with the quantum capacities in the
presence of free backward or two-way public classical communication. These
capacities are proved continuous on the interior of the set of non-zero
capacity channels by considering mutual simulation between similar channels.Comment: 12 pages, Revised according to referee's suggestion
Continuity of the Maximum-Entropy Inference
We study the inverse problem of inferring the state of a finite-level quantum
system from expected values of a fixed set of observables, by maximizing a
continuous ranking function. We have proved earlier that the maximum-entropy
inference can be a discontinuous map from the convex set of expected values to
the convex set of states because the image contains states of reduced support,
while this map restricts to a smooth parametrization of a Gibbsian family of
fully supported states. Here we prove for arbitrary ranking functions that the
inference is continuous up to boundary points. This follows from a continuity
condition in terms of the openness of the restricted linear map from states to
their expected values. The openness condition shows also that ranking functions
with a discontinuous inference are typical. Moreover it shows that the
inference is continuous in the restriction to any polytope which implies that a
discontinuity belongs to the quantum domain of non-commutative observables and
that a geodesic closure of a Gibbsian family equals the set of maximum-entropy
states. We discuss eight descriptions of the set of maximum-entropy states with
proofs of accuracy and an analysis of deviations.Comment: 34 pages, 1 figur