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arXiv:quant-ph/0308012 · Physical Review Letters

Classical Capacity of the Lossy Bosonic Channel: The Exact Solution

Vittorio Giovannetti, Saikat Guha, Seth Lloyd, Lorenzo Maccone, Jeffrey H. Shapiro, Horace P. Yuen

Published Jan 15, 2004• 341 citations
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Abstract

The classical capacity of the lossy bosonic channel is calculated exactly. It is shown that its Holevo information is not superadditive, and that a coherent-state encoding achieves capacity. The capacity of far-field, free-space optical communications is given as an example.
DOI
10.1103/physrevlett.92.027902

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The capacity C for transmitting classical information is investigated for Bosonic channels with isotropic Gaussian noise. For the pure‐loss channel—in which signal photons may be lost in propagation—the exact value of C is derived. The Holevo information of this channel is shown to be additive, and a “classical” encoding procedure employing coherent states is shown to achieve capacity. For active channel models—in which noise photons are injected from an external environment or the signal is amplified with unavoidable quantum noise—upper and lower bounds are obtained for the capacity. These bounds are asymptotically tight at low and high noise levels. Exact capacity results—given by the lower bounds—would follow from proving the conjecture that a coherent‐state input minimizes the output entropy from such channels.
Optimization methods aimed at estimating the capacities of a general Gaussian channel are developed. Specifically evaluation of classical capacity as maximum of the Holevo information is pursued over all possible Gaussian encodings for the lossy bosonic channel, but extension to other capacities and other Gaussian channels seems feasible. Solutions for both memoryless and memory channels are presented. It is first dealt with single-use (single-mode) channel where the capacity dependence on channel's parameters is analyzed providing the full classification of possible cases. Then, it is dealt with multiple uses (multimode) channel where the capacity dependence on the (multimode) environment state is analyzed when both total environment energy and environment purity are fixed. This allows a fair comparison among different environments, thus understanding the role of memory (intermode correlations) and phenomenon like superadditivity of the capacity. The developed methods are also used for deriving transmission rates with heterodyne and homodyne measurements at the channel output. Classical capacity and transmission rates are presented within a unique framework where the rates can be treated as logarithmic approximations of the capacity.
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