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arXiv:1408.3255 · Theory of Probability and Its Applications

Coding Theorems for Hybrid Channels. II

A. A. Kuznetsova, Alexander Semenovich Holevo

Published Jan 1, 2015• 5 citations
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Abstract

The present paper continues investigation of the capacities of measurement (quantum-classical) channels in the most general setting, initiated in [A. A. Kuznetsova and A. S. Holevo, Theory Probab. Appl., 58 (2014), pp. 264--285]. The proof of coding theorems is given for the classical capacity and entanglement-assisted classical capacity of the measurement channel with arbitrary output alphabet, without assuming that the channel is given by a bounded operator-valued density.

DOI
10.1137/s0040585x97986977

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The present paper continues investigation of the capacities of measurement (quantum-classical) channels in the most general setting, initiated in [A. A. Kuznetsova and A. S. Holevo, Theory Probab. Appl., 58 (2014), pp. 264--285]. The proof of coding theorems is given for the classical capacity and entanglement-assisted classical capacity of the measurement channel with arbitrary output alphabet, without assuming that the channel is given by a bounded operator-valued density.
The paper of Hsieh, Luo, and Brun (HLB) [Phys. Rev. A 78, 042306 (2008)] contains several issues with the capacity theorem presented there, one of which is the suggestion that a sender and receiver can achieve entanglement-assisted classical capacity without any entanglement at all, and another of which is a violation of the Holevo bound. There is also an issue with the converse proof of the capacity theorem. In this comment, I point out these issues and provide revisions of the capacity theorem and the converse proof.
In this thesis we analyse the type of states and ensembles which achieve the capacity for certain quantum channels carrying classical information. We first concentrate on the product-state capacity of a particular quantum channel, that is, the capacity which is achieved by encoding the output states from a source into codewords comprised of states taken from ensembles of non-entangled states and sending them over copies of the quantum channel. Using the "single-letter" formula proved independently by Holevo and by Schumacher and Westmoreland we obtain the product-state capacity of the qubit quantum amplitude-damping channel, which is determined by a transcendental equation in a single real variable and can be solved numerically. We demonstrate that the product-state capacity of this channel can be achieved using a minimal ensemble of non-orthogonal pure states. Next we consider the classical capacity of two quantum channels with memory, namely a periodic channel with quantum depolarising channel branches and a convex combination of quantum channels. We prove that the classical capacity for each of the classical memory channels mentioned above is, in fact, equal to the respective product-state capacities. For those channels this means that the classical capacity is achieved without the use of entangled input-states. Next we introduce the channel coding theorem for memoryless quantum channels, providing a known proof by Winter for the strong converse of the theorem. We then consider the strong converse to the channel coding theorem for a periodic quantum channel.
Unextendibility of quantum states and channels is inextricably linked to the no-cloning theorem of quantum mechanics, it has played an important role in understanding and quantifying entanglement, and more recently it has found applications in providing limitations on quantum error correction and entanglement distillation. Here we generalize the framework of unextendibility to quantum measurements and define kkk-extendible measurements for every integer k≥2k\ge 2k≥2. Our definition provides a hierarchy of semidefinite constraints that specify a set of measurements containing every measurement that can be realized by local operations and one-way classical communication. Furthermore, the set of kkk-extendible measurements converges to the set of measurements that can be realized by local operations and one-way classical communication as k→∞k\to \inftyk→∞. To illustrate the utility of kkk-extendible measurements, we establish a semidefinite programming upper bound on the one-shot classical capacity of a channel, which outperforms the best known efficiently computable bound from [Matthews and Wehner, IEEE Trans. Inf. Theory 60, pp. 7317-7329 (2014)] and also leads to efficiently computable upper bounds on the nnn-shot classical capacity of a channel.
A coding theorem for entanglement-assisted communication via an infinite-dimensional quantum channel with linear constraints is extended to a natural degree of generality. Relations between the entanglement-assisted classical capacity and χ-capacity of constrained channels are obtained, and conditions for their coincidence are given. Sufficient conditions for continuity of the entanglement-assisted classical capacity as a function of a channel are obtained. Some applications of the obtained results to analysis of Gaussian channels are considered. A general (continuous) version of the fundamental relation between coherent information and the measure of privacy of classical information transmission via an infinite-dimensional quantum channel is proved.