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IEEE International Symposium on Information Theory (ISIT)

Information capacity of bosonic channels

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

Published Dec 23, 2004• 3 citations
Publisher

Abstract

The capacity C for transmitting classical information is investigated for noisy bosonic channel models. An exact result is obtained for the pure-loss case. Upper and lower bounds are established for channels with active noise sources
DOI
10.1109/isit.2004.1365362

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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.
The classical capacity of a noisy optical channel is the maximum rate at which information can be transmitted reliably, given a constraint on the mean input energy. For the ubiquitous bosonic Gaussian channels, which include standard models of loss and noise in fibre and free-space optical links, determining this rate requires addressing whether entanglement between channel uses or non-Gaussian signals can outperform Gaussian encodings. More precisely, the question is whether the energy-constrained Holevo information is additive across arbitrarily many channel uses and coincides with its Gaussian counterpart. We answer these fundamental questions in the affirmative for every single-mode bosonic Gaussian channel at every input energy. Namely, we prove that independently prepared Gaussian states with Gaussian-distributed random displacements achieve the capacity, including in the low-energy regime where squeezing the signals competes with encoding information in their displacement. Our proof connects an entanglement inequality to a sharp, additive lower bound on output entropy. We then characterise communication above capacity: under a strict total photon-number cutoff, the probability of correct decoding decreases exponentially with the number of channel uses, and we derive explicit bounds on the decay exponent. When only the mean photon number is constrained, a simple energy-rescaling formula determines exactly the entire asymptotic rate--error trade-off: the capacity at mean photon number NNN and allowed average decoding error εεε equals the vanishing-error capacity at mean photon number N/(1−ε)N/(1-ε)N/(1−ε). Together, these results identify the ultimate limits of classical communication for some of the most fundamental models of optical communication.
The maximum rates for information transmission through noisy quantum channels has primarily been developed for memoryless channels, where the noise on each transmitted state is treated as independent. Many real-world communication channels experience noise which is modeled better by errors that are correlated between separate channel uses. In this Brief Report, upper bounds on the classical information capacities of a class of quantum memory channels are derived. The class of channels consists of indecomposable quantum memory channels, a generalization of classical indecomposable finite-state channels.
An upper limit is given to the amount of quantum information that can be transmitted reliably down a noisy, decoherent quantum channel using the high-probability states of quantum sources. A class of quantum error-correcting codes is presented that allows the information transmitted to attain this limit. The result is a quantum analog of Shannon's bound and code for the noisy classical channel [C. E. Shannon and W. Weaver, The Mathematical Theory of Communication (University of Illinois Press, Chicago, 1948)].
We consider a line with noise in the simplest case. Loss does not add noise. Amplification via phase insensitive amplifiers do add noise. A lower bound of this capacity is the quantum analog to the Shannon capacity of a linear channel with additive white Gaussian noise, namely the difference of the Von Neumann entropy of the signal plus noise at the output of the line and the entropy of the noise alone. We show that this expression is indeed the capacity for the case of an amplifier with infinitesimal gain G=1+εG = 1+εG=1+ε, and for a cascade of an amplifier with arbitrary gain and a large loss, such that the overall gain of the cascade is infinitesimal.