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Nature Communications

Regeneration limit of classical Shannon capacity

Mariia A. Sorokina, Sergei K. Turitsyn

Published May 20, 2014• 74 citations
Publisher

Abstract

Since Shannon derived the seminal formula for the capacity of the additive linear white Gaussian noise channel, it has commonly been interpreted as the ultimate limit of error-free information transmission rate. However, the capacity above the corresponding linear channel limit can be achieved when noise is suppressed using nonlinear elements; that is, the regenerative function not available in linear systems. Regeneration is a fundamental concept that extends from biology to optical communications. All-optical regeneration of coherent signal has attracted particular attention. Surprisingly, the quantitative impact of regeneration on the Shannon capacity has remained unstudied. Here we propose a new method of designing regenerative transmission systems with capacity that is higher than the corresponding linear channel, and illustrate it by proposing application of the Fourier transform for efficient regeneration of multilevel multidimensional signals. The regenerative Shannon limit—the upper bound of regeneration efficiency—is derived. The Shannon limit describes the limit of error-free information transmission and thus the information that can be transmitted in telecommunications. Here, the authors derive the Shannon limit for nonlinear, regenerative systems, expanding on the classical linear case.

DOI
10.1038/ncomms4861

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Since Shannon derived the seminal formula for the capacity of the additive linear white Gaussian noise channel, it has commonly been interpreted as the ultimate limit of error-free information transmission rate. However, the capacity above the corresponding linear channel limit can be achieved when noise is suppressed using nonlinear elements; that is, the regenerative function not available in linear systems. Regeneration is a fundamental concept that extends from biology to optical communications. All-optical regeneration of coherent signal has attracted particular attention. Surprisingly, the quantitative impact of regeneration on the Shannon capacity has remained unstudied. Here we propose a new method of designing regenerative transmission systems with capacity that is higher than the corresponding linear channel, and illustrate it by proposing application of the Fourier transform for efficient regeneration of multilevel multidimensional signals. The regenerative Shannon limit—the upper bound of regeneration efficiency—is derived. The Shannon limit describes the limit of error-free information transmission and thus the information that can be transmitted in telecommunications. Here, the authors derive the Shannon limit for nonlinear, regenerative systems, expanding on the classical linear case.
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.
edu Abstract: Wedevelop information-theoretical description ofnonlinear optical communication systems. We present a non-perturbative calculation oftheinformation capacity fordifferent modulation formats, anddemonstrate thatfiber nonlinearity leads toexponential suppression ofthechannel capacity. © 2004Optical Society ofAmerica OCIScodes:(060.2330) Fiberoptics communications; (190.4370) Nonlinear optics, fibers; (190.4380) Nonlinear optics, four-wave mixing Theperformance ofanycommunication system isultimately limited bythesignal tonoise ratio ofthe received signal andavailable bandwidth. Thislimitation canbestated moreformally byusing theconcept of information capacity originally introduced within theframework ofinformation theory (1). Theinformation capacity isdefined asthemaximumpossible bitrateforerror-free transmission inthepresence ofnoise. For alinear communication channel withadditive noise, andatotal signal powerconstraint attheinput, the capacity isgiven bythecelebrated Shannon formula (1)
This tutorial reviews the Holevo capacity limit as a universal tool to analyze the ultimate transmission rates in a variety of optical communication scenarios, ranging from conventional optically amplified fiber links to free-space communication with power-limited optical signals. The canonical additive white Gaussian noise model is used to describe the propagation of the optical signal. The Holevo limit exceeds substantially the standard Shannon limit when the power spectral density of noise acquired in the course of propagation is small compared to the energy of a single photon at the carrier frequency per unit time-bandwidth area. General results are illustrated with a discussion of efficient communication strategies in the photon-starved regime.
We analyze the information-capacity limitations of the optical communication channel, as determined by noise accumulation from optical amplification and nonlinear wave-mixing. We review the concepts of signal-to-noise ratio and entropy for binary-coded and continuous communications, leading to a definition of ultimate capacity for the optically-amplified channel. A unified quantum model, describing both amplification and nonlinearity limitations, makes possible to determine the power transmission window within which the channel capacity can be maximized.
Fiber-optic communications are moving to coherent detection in order to increase their spectral efficiency, i.e., their channel capacity per unit bandwidth. At power levels below the threshold for significant nonlinear effects, the channel model for such operation a linear time-invariant filter followed by additive Gaussian noise is one whose channel capacity is well known from Shannon's noisy channel coding theorem. The fiber channel, however, is really a bosonic channel, meaning that its ultimate classical information capacity must be determined from quantum-mechanical analysis, viz. from the Holevo-Schumacher-Westmoreland (HSW) theorem. Based on recent results establishing the HSW capacity of a linear (lossy or amplifying) channel with additive Gaussian noise, we provide a general continuous-time result, namely the HSW capacity of a linear time-invariant (LTI) bosonic channel with additive Gaussian noise arising from a thermal environment. In particular, we treat quasi-monochromatic communication under an average power constraint through a channel comprised of a stable LTI filter that may be attenuating at all frequencies or amplifying at some frequencies and attenuating at others. Phase-insensitive additive Gaussian noise-associated with the continuous-time Langevin noise operator needed to preserve free-field commutator brackets is included at the filter output. We compare the resulting spectral efficiencies with corresponding results for heterodyne and homodyne detection over the same channel to assess the increased spectral efficiency that might be realized with optimum quantum reception.
The classical capacity of quantum channels is the tight upper bound for the transmission rate of classical information. This is a quantum counterpart of the foundational notion of the channel capacity introduced by Shannon. Bosonic Gaussian quantum channels provide a good model for optical communications. In order to properly define the classical capacity for these quantum systems, an energy constraint at the channel input is necessary, as in the classical case. A further restriction to Gaussian input ensembles defines the Gaussian (classical) capacity, which can be studied analytically. It also provides a lower bound on the classical capacity and moreover, it is conjectured to coincide with the classical capacity. Therefore, the Gaussian capacity is a useful and important notion in quantum information theory. Recently, we have shown that the study of both the classical and Gaussian capacity of an arbitrary single mode Gaussian quantum channel can be reduced to the study of a particular fiducial channel. In this work we consider the Gaussian capacity of the fiducial channel, discuss its additivity and analyze its dependence on the channel parameters. In addition, we extend previously obtained results on the optimal channel environment to the single mode fiducial channel. In particular, we show that the optimal channel environment for the lossy, amplification, and phase conjugating channels is given by a pure quantum state if its energy is constrained.