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arXiv:0903.1638 · IEEE Transactions on Information Theory

An Infinite Sequence of Additive Channels: The Classical Capacity of Cloning Channels

Kamil Brádler

Published Aug 1, 2011• 33 citations
arXivHTMLPDF

Abstract

We introduce an infinite sequence of quantum channels for which the Holevo capacity is additive. The channel series is closely related to the quantum channels arising from universal quantum cloning machines. The additivity proof is motivated by a special property the studied channels enjoy: the property of conjugate degradability. As a consequence of the announced proof, we also provide an easy way of proving the additivity of the Holevo capacity for the original Unruh channel for which the quantum capacity is already known. Consequently, we present not only an infinite series of finite-dimensional channels but also a nontrivial example of an infinite-dimensional channel for which the classical and quantum channel capacities are easily calculable.

DOI
10.1109/tit.2011.2158896

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Theory of Probability and Its Applications•Jan 1, 2008•15 citations
We introduce an infinite sequence of quantum channels for which the Holevo capacity is additive. The channel series is closely related to the quantum channels arising from universal quantum cloning machines. The additivity proof is motivated by a special property the studied channels enjoy: the property of conjugate degradability. As a consequence of the announced proof, we also provide an easy way of proving the additivity of the Holevo capacity for the original Unruh channel for which the quantum capacity is already known. Consequently, we present not only an infinite series of finite-dimensional channels but also a nontrivial example of an infinite-dimensional channel for which the classical and quantum channel capacities are easily calculable.
The notion of the Holevo capacity for arbitrarily constrained infinite dimensional quantum channels is introduced. It is shown that despite nonexistence of an optimal ensemble in this case it is possible to define the notion of the output optimal average state for such a channel. The characterization of the output optimal average state and a minimax expression for the Holevo capacity are obtained. This makes it possible to prove equivalence of several additivity properties for infinite dimensional quantum channels. The notion of the χ\chiχ-function for an infinite dimensional channel is considered, its strong concavity and lower semicontinuity are shown. The problem of continuity of the Holevo capacity is also discussed. It is shown that the Holevo capacity is continuous function of a channel in the finite dimensional case while in general it is only lower semicontinuous. This conclusion is confirmed by the example. The main result of this note is the statement that additivity of the Holevo capacity for all finite dimensional channels implies additivity of the Holevo capacity for all infinite dimensional channels with arbitrary constraints. The subadditivity of the χ\chiχ-function for two infinite dimensional channels with one of them noiseless or entanglement breaking is also proved.
A quantum channel is conjugate degradable if the channel’s environment can be simulated up to complex conjugation using the channel’s output. For all such channels, the quantum capacity can be evaluated using a single-letter formula. In this article we introduce conjugate degradability and establish a number of its basic properties. We then use it to calculate the quantum capacity of N to N+1 and 1 to M universal quantum cloning machines as well as the quantum capacity of a channel that arises naturally when data are being transmitted to an accelerating receiver. All the channels considered turn out to have strictly positive quantum capacity, meaning they could be used as part of a communication system to send quantum states reliably.
Conjugate degradable channels are channels whose quantum capacity is calculable. They were defined and studied in [1] where, however, only channels that are both degradable and conjugate degradable were found. In this paper we bring the very first example of conjugate degradable channels that are not degradable. We also identify the physical origin of these channels and show that they belong to the class of optimal universal asymmetric cloners. We thus not only positively answer the question whether conjugate degradable channels form a new class of channels with a single-letter capacity formula but as a side result we also calculate the quantum capacity of the optimal asymmetric 1->1+1 cloning channels.
Abstract Finding the classical capacity of a quantum channel is not easy, yet we are able to analytically calculate this capacity for new channels. We analyze the bounds of the Holevo capacity and classical capacity for the generalized Pauli channels. In particular, by generalizing earlier results for the Weyl channels, we obtain the lower and upper bounds of the Holevo capacity and show that, if these bounds coincide, the Holevo capacity is weakly additive. We also prove the weak additivity of the lower bound. Two new examples of the generalized Pauli channels with known classical capacity are presented. Finally, we relate the change rate of the classical capacity to the P-divisibility of Pauli channels.
We study quantum channels that are close to another channel with weakly additive Holevo information, and we derive upper bounds on their classical capacity. Examples of channels with weakly additive Holevo information are entanglement-breaking channels, unital qubit channels, and Hadamard channels. Related to the method of approximate degradability, we define approximation parameters for each class above, which measure how close an arbitrary channel is to satisfying the respective property. This gives us upper bounds on the classical capacity in terms of functions of the approximation parameters, as well as an outer bound on the dynamic capacity region of a quantum channel. Since these parameters are defined in terms of the diamond distance, the upper bounds can be computed efficiently using semidefinite programming (SDP). We exhibit the usefulness of our method with two example channels: a convex mixture of amplitude damping and depolarizing noise and a composition of amplitude damping and dephasing noise. For both channels, our bounds perform well in certain regimes of the noise parameters in comparison to a recently derived SDP upper bound on the classical capacity. Along the way, we define the notion of a generalized channel divergence (which includes the diamond distance as an example), and we prove that for jointly covariant channels these quantities are maximized by purifications of a state invariant under the covariance group. This latter result may be of independent interest.
This paper is devoted to further study of the Holevo capacity of infinite-dimensional quantum channels. The existence of a unique optimal average state for a quantum channel constrained by an arbitrary convex set of states is shown. The minimax expression for the Holevo capacity of a constrained channel is obtained. The χ\chiχ-function and the convex closure of the output entropy of an infinite-dimensional quantum channel are considered. It is shown that the χ\chiχ-function of an arbitrary channel is lower semicontinuous on the set of all states and has continuous restrictions to subsets of states with continuous output entropy. The explicit expression for the convex closure of the output entropy of an infinite-dimensional quantum channel is obtained and its properties are explored. It is shown that the convex closure of the output entropy coincides with the convex hull of the output entropy on the set of states with finite output entropy and, similarly to the χ\chiχ-function, it has continuous restrictions to subsets of states with continuous output entropy. The applications of the obtained results to the theory of entanglement are considered. The properties of the convex closure of the output entropy make it possible to generalize some results related to the additivity problem to the infinite-dimensional case.