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Journal of Physics Conference Series

Calculation of mutual information for nonlinear optical fiber communication channel at large SNR within path-integral formalism

А. В. Резниченко, Ivan Sergeevich Terekhov, Sergei K. Turitsyn

Published Apr 20, 2017• 5 citations
Publisher

Abstract

Using the path-integral technique we calculate the mutual information for the fiber optical channel modelled by the nonlinear Schrödinger equation with additive Gaussian noise. At large signal-to-noise ratio (SNR) we present the mutual information through the path-integral which is convenient for the perturbative expansion both in nonlinearity and dispersion. In the leading order in 1/SNR we demonstrate that the mutual information is determined through the averaged logarithm of the normalization factor Λ of the conditional probability density function P [ Y | X ]. In the limit of small noise and small nonlinearity we derive analytically the first nonzero nonlinear correction to the mutual information for the channel. For the arbitrary nonlinearity we restrict the mutual information by the low bound obtained from the Jensen's inequality and analyze the bound for the case of large dispersion.

DOI
10.1088/1742-6596/826/1/012026

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Using the path-integral technique we calculate the mutual information for the fiber optical channel modelled by the nonlinear Schrödinger equation with additive Gaussian noise. At large signal-to-noise ratio (SNR) we present the mutual information through the path-integral which is convenient for the perturbative expansion both in nonlinearity and dispersion. In the leading order in 1/SNR we demonstrate that the mutual information is determined through the averaged logarithm of the normalization factor Λ of the conditional probability density function P [ Y | X ]. In the limit of small noise and small nonlinearity we derive analytically the first nonzero nonlinear correction to the mutual information for the channel. For the arbitrary nonlinearity we restrict the mutual information by the low bound obtained from the Jensen's inequality and analyze the bound for the case of large dispersion.
We consider the information channel described by the Schrödinger equation with additive Gaussian noise. We introduce the model of the input signal and the model of the output signal receiver. For this channel, using perturbation theory for the small nonlinearity parameter, we calculate the first three terms of the expansion of the conditional probability density function in the nonlinearity parameter. At a large signal-to-noise power ratio, we calculate the conditional entropy, output signal entropy, and mutual information in the leading and next-to-leading order in the nonlinearity parameter and in the leading order in the parameter 1/SNR. Using the mutual information, we find the optimal input signal distribution and channel capacity in the leading and next-to-leading order in the nonlinearity parameter. Finally, we present the method of the construction of the input signal with the optimal statistics for the given shape of the signal.
We have studied the squeezing of the two-mode radiation fields in a nonlinear optical fiber with gain. The coupling nonlinear Schrödinger equations corresponding to the circularly polarized radiation fields with gain are derived by means of the method of path integrals. The minimized power spectra of the quantum noise of these fields are obtained with a first-order perturbation approximation in the frequency domain. Detailed numerical analyses of the results are performed and presented graphically, taking zero-dispersion, normal and anomalous dispersion into account.
Multi-soliton pulses are studied under small perturbations. By means of mutual information, we show that the nonlinear spectrum becomes correlated when a 64-PSK modulated 2-eigenvalues soliton is perturbed by a small additive white Gaussian noise (AWGN). Transmission over two channels are considered: an AWGN channel (back-to-back scenario) and an ideal optical fiber link with distributed noise. We numerically show that the Jost coefficients b(λk) are less correlated than the discrete spectral amplitudes qd(λk) while either of them can be alternatively used for modulation. In both channels, we compare the achievable information rates (AIRs) of joint detection and separate detection if the modulation is over either qd(λk) or b(λk). We show that if the length of the link is relatively short, the separate detection of b(λk) gives almost the same total information rate as the joint detection of b(λk) or qd(λk). Moreover, the AIRs are compared with one of a first-order soliton with the same modulation.
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.
We analyze the quantum dynamics of radiation propagating in a single-mode optical fiber with dispersion, nonlinearity, and Raman coupling to thermal phonons. We start from a fundamental Hamiltonian that includes the principal known nonlinear effects and quantum-noise sources, including linear gain and loss. Both Markovian and frequency-dependent, non-Markovian reservoirs are treated. This treatment allows quantum Langevin equations, which have a classical form except for additional quantum-noise terms, to be calculated. In practical calculations, it is more useful to transform to Wigner or +P quasi-probability operator representations. These transformations result in stochastic equations that can be analyzed by use of perturbation theory or exact numerical techniques. The results have applications to fiber-optics communications, networking, and sensor technology.
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.