This paper is devoted to the study of χ-capacity, closely related to the classical capacity of infinite-dimensional quantum channels. For such channels generalized ensembles are defined as probability measures on the set of all quantum states. We establish the compactness of the set of generalized ensembles with averages in an arbitrary compact subset of states. This result enables us to obtain a sufficient condition for the existence of the optimal generalized ensemble for an infinite-dimensional channel with input constraint. This condition is shown to be fulfilled for Bosonic Gaussian channels with constrained mean energy. In the case of convex constraints, a characterization of the optimal generalized ensemble extending the "maximal distance property" is obtained.1. Introduction. This paper is devoted to the systematic study of the classical capacity (more precisely, a closely related quantity-the χ-capacity [6]) of infinitedimensional quantum channels, following [8], [10], [17]. While major attention in quantum information theory up to now has been paid to finite-dimensional systems, there is an important and interesting class of Gaussian channels (see, e.g., [9], [4], [16]) which act in infinite-dimensional Hilbert space. Although many problems of Gaussian bosonic systems with a finite number of modes can be solved with finite-dimensional matrix techniques, a general underlying Hilbert space operator analysis is indispensable.Moreover, it was observed recently [17] that Shor's famous proof of the global equivalence of different forms of the additivity conjecture is related to the weird discontinuity of the χ-capacity in the infinite-dimensional case. All this calls for a mathematically rigorous treatment involving specific results from the operator theory in Hilbert space and measure theory.There are two important features essential for channels in infinite dimensions. One is the necessity of the input constraints (such as the mean energy constraint for Gaussian channels) to prevent infinite capacities (although considering input constraints was recently shown to be quite useful also in the study of the additivity conjecture for channels in finite dimensions [10]). Another is the natural appearance of infinite, and, in general, "continuous" state ensembles understood as probability measures on the set of all quantum states. By using compactness criteria from probability theory and operator theory we can show that the set of all generalized ensembles with the average in a compact set of states is itself a compact subset of the set of all probability measures. With this in hand we give a sufficient condition for the existence of an optimal generalized ensemble for a constrained quantum channel. This condition can be

Several equivalent formulations of the additivity conjecture for constrained channels, which formally is substantially stronger than the unconstrained additivity, are given. 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. 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.Running title: Shor's channel extension and constrained channels *

1 It is also used the term "lower envelope" [4].

We consider the family of energy-constrained diamond norms on the set of Hermitian-preserving linear maps (superoperators) between Banach spaces of trace class operators. We prove that any norm from this family generates the strong (pointwise) convergence on the set of all quantum channels (which is more adequate for describing variations of infinite-dimensional channels than the diamond norm topology).We obtain continuity bounds for information characteristics (in particular, classical capacities) of energy-constrained quantum channels (as functions of a channel) with respect to the energy-constrained diamond norms which imply uniform continuity of these characteristics with respect to the strong convergence topology.

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In this paper we describe the recent changes to the curriculum of the second year practical laboratory course in the School of Physics and Astronomy at the University of Nottingham. In particular, we describe how Matlab has been implemented as a teaching tool and discuss both its pedagogical advantages and disadvantages in teaching undergraduate students about computer interfacing and instrument control techniques. We also discuss the motivation for converting the interfacing language that is used in the laboratory from LabView to Matlab. We describe an example of a typical experiment the students are required to complete and we conclude by briefly assessing how the recent curriculum changes have affected both student performance and compliance.

We start with Fannes' type and Winter's type tight continuity bounds for the quantum conditional mutual information and their specifications for states of special types.