We consider a model nondispersive nonlinear optical fiber channel with an additive Gaussian noise. Using Feynman path-integral technique, we find the optimal input signal distribution maximizing the channel's per-sample mutual information at large signal-to-noise ratio in the intermediate power range. The optimal input signal distribution allows us to improve previously known estimates for the channel capacity. We calculate the output signal entropy, conditional entropy, and per-sample mutual information for Gaussian, half-Gaussian, and modified Gaussian input signal distributions. We demonstrate that in the intermediate power range the capacity (the per-sample mutual information for the optimal input signal distribution) is greater than the per-sample mutual information for half-Gaussian input signal distribution considered previously as the optimal one. We also show that the capacity grows as loglogP in the intermediate power range, where P is the signal power.

Understanding the properties of quantum matter is an outstanding challenge in science. In this work, we demonstrate how machine learning methods can be successfully applied for the classification of various regimes in single-particle and many-body systems. We realize neural network algorithms that perform a classification between regular and chaotic behavior in quantum billiard models with remarkably high accuracy. By taking this method further, we show that machine learning techniques allow to pin down the transition from integrability to many-body quantum chaos in Heisenberg XXZ spin chains. Our results pave the way for exploring the power of machine learning tools for revealing exotic phenomena in complex quantum many-body systems.Introduction. Significant attention to machine learning techniques is related to their applications in tasks of finding patterns in data, such as image recognition, speech analysis, computer vision, and many other domains [1]. Quantum physics is well known to produce atypical patterns in data, which are in principle can be revealed using machine learning methods [2]. This idea has stimulated an intensive ongoing research of this subject. The scope so far includes identification of phases of quantum matter and detecting phase transitions [4][5][6][7][8][9][10][11], as well as representing quantum states of many-body systems in regimes that are intractable for existing exact numerical approaches [15][16][17]. Another branch of research is related to the applications of machine learning tools to the analysis of experimental data [18][19][20]. Recently, a machine learning approach has been used for processing data from gas microscopes and evaluating predictions of competing theories that describe the doped Hubbard model without a bias towards one of the theories [21].Remarkable progress on building large-scale quantum simulators has opened fascinating prospects for the observation of novel quantum phases and exotic states [22][23][24][25]. They also provide interesting insights to traditionally challenging problems in studies of complex quantum systems, such as investigation of quantum critical dynamics and quantum chaos [26]. Quantum systems with chaotic behaviour are of great interest in the view of a possibility to explore quantum scars in them [27]. Quantum many-body scars can be potentially compatible with long-lived states, which are of importance for quantum information processing. A standard criterion for the separation between regular and chaotic regimes is based on the nearest-neighbor energy level statistics [28,29]: Poisson and Wigner-Dyson distributions correspond to integrable and chaotic systems, respectively. However, the energy level statistics of highly excited states is not always directly accessible in experiments.From the machine learning perspective, an interesting problem is to understand whereas it is possible to distinguish between regular and chaotic behavior, in the best-