The capacity of the bosonic channel with additive Gaussian noise is unknown, but there is a known lower bound that is conjectured to be the capacity. We have quantified the gap that exists between this known achievable rate and rates achievable by the known methods of detection including direct, heterodyne, and homodyne detection. We have also quantified these capacities in the case of multiple independent spatial modes in terms of spectral and photon efficiency. Furthermore, we have considered the ergodic and outage capacities of fad-ing channel models for far-field and near-field propagation through atmospheric turbulence. For the far field, good models for the transmissivity statistics are known. For the near field we establish bounds on these capacities, and we show that these bounds are reasonably tight. Finally, we extend the results for ergodic capacity to the case of multiple spatial modes where a turbulent atmosphere results in crosstalk between different spatial modes.
The capacity of the bosonic channel with additive Gaussian noise is unknown, but there is a known lower bound that is conjectured to be the capacity. We have quantified the gap that exists between this known achievable rate and rates achievable by the known methods of detection including direct, heterodyne, and homodyne detection. We have also quantified these capacities in the case of multiple independent spatial modes in terms of spectral and photon efficiency. Furthermore, we have considered the ergodic and outage capacities of fad-ing channel models for far-field and near-field propagation through atmospheric turbulence. For the far field, good models for the transmissivity statistics are known. For the near field we establish bounds on these capacities, and we show that these bounds are reasonably tight. Finally, we extend the results for ergodic capacity to the case of multiple spatial modes where a turbulent atmosphere results in crosstalk between different spatial modes.
Vacuum-propagation optical communication with high photon efficiency (many bits/photon) and high spectral efficiency (many bits/s ·Hz) requires operation in the near-field power transfer regime with a large number of spatial modes. For terrestrial propagation paths, however, the effects of atmospheric turbulence must be factored into the photon and spectral efficiency assessments. In Part I of this study [N. Chandrasekaran and J. H. Shapiro, “Photon Information Efficient Communication Through Atmospheric Turbulence-Part I: Channel Model and Propagation Statistics,” J. Lightw. Technol., vol. 32, no. 6, pp. 1075-1087, Mar. 2014], modal-transmissivity statistics were derived for the turbulent channel that depend solely on the mutual coherence function of the atmospheric Green's function, and these bounds were evaluated for ~ 200 spatial-mode systems whose transmitters used either focused-beam (FB), Hermite-Gaussian (HG), or Laguerre-Gaussian (LG) modes and whose receivers either did or did not employ adaptive optics. This Part II paper derives upper and lower bounds for the ergodic Holevo capacities of classical and private information transmission over the multiple spatial-mode turbulent channel that can be evaluated from Part I's transmissivity statistics. Also included are bounds on the ergodic capacity for on-off keying encoding and direct detection. It is shown that: 1) adaptive optics are not necessary to realize high photon information efficiency and high spectral efficiency simultaneously; 2) an FB-mode system with perfect adaptive optics outperforms its HG-mode and LG-mode counterparts; and 3) the converse is true when adaptive optics are not employed.
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.
We prove an upper bound on the private capacity of the single-mode noiseless bosonic wiretap channel. Combined with a previous lower bound, we obtain the low photon-number asymptotic expression for the private capacity. We then show that the multiple-mode noiseless bosonic wiretap channel is equivalent to parallel single-mode channels, hence the single-mode bounds can be applied. Finally, we consider multiple-spatial-mode propagation through atmospheric turbulence, and derive a private-capacity lower bound that only requires second moments of the channel matrix.
Optical communication with high photon-efficiency (many bits/photon) and high spectral efficiency (SE) (many bits/s-Hz) cannot be achieved unless multiple spatial modes are employed. For vacuum propagation, it is known that achieving 10 bits/photon and 5 bits/s-Hz requires 189 low-loss spatial modes at the ultimate Holevo limit and 4500 such modes at the Shannon limit for on-off keying with direct detection. For terrestrial propagation paths, however, atmospheric turbulence corrupts multiple spatial-mode operation. This paper derives power-transmissivity bounds and average intermodal crosstalks for the turbulent channel that depend solely on the mutual coherence function of the atmospheric Green's function. These statistics are then evaluated for ~ 200 spatial-mode systems whose transmitters use either focused-beam, Hermite-Gaussian (HG), or Laguerre-Gaussian (LG) modes and whose receivers either do or do not employ adaptive optics. It is shown that: (1) adaptive optics are not necessary for achieving both high photon information efficiency (PIE) and high SE; (2) systems employing HG or LG modes achieve the same capacities through turbulence; and (3) the orbital angular momentum carried by LG modes does not provide turbulence immunity. In the companion paper [N. Chandrasekaran, J. H. Shapiro, and L. Wang, “Photon Information Efficient Communication Through Atmospheric Turbulence-Part II: Bounds on Ergodic Classical and Private Capacities,” J. Lightw. Technol., vol. 32, no. 6, pp. 1088-1097, Mar. 2014], the transmissivity bounds are used to quantify the turbulence-induced loss in PIE versus SE performance for these mode sets.
In this letter, we analyze the average probability of error of quantum modulation schemes in weakly turbulent quantum communication channels utilizing coherent states. Atmospheric turbulence-induced fading is a significant challenge that degrades the performance of free-space quantum communication systems. We derive and present closed-form expressions for the average probability of error for various detection methods: classical (shot-noise limited) single-photon detection, coherent detection, and the optimal quantum hypothesis-testing (Helstrom) detection for binary quantum modulation schemes. Our analysis focuses on the transmission of classical data modulated by quantum coherent states over a log-normal turbulent channel. Additionally, we discuss the lower bounds of the average capacities for Helstrom detection, providing comprehensive insights into the achievable information rates in realistic atmospheric conditions. The derived expressions show excellent agreement with numerical Monte Carlo simulations across a broad range of average transmitted photon numbers, offering valuable guidelines for the design and optimization of quantum communication systems.
Semiclassical models for multiple-user optical communication cannot assess the ultimate limits on reliable communication as permitted by the laws of physics. In all optical communications settings that have been analyzed within a quantum framework so far, the gaps between the quantum limit to the capacity and the Shannon limit for structured receivers become most significant in the low photon-number regime. Here, we present a quantum treatment of a multiple-transmitter multiple-receiver multi-spatial-mode free-space interference channel with diffraction-limited loss and a thermal background. We consider the performance of a laser-light (coherent state) encoding in conjunction with various detection strategies such as homodyne, heterodyne, and joint detection. Joint detection outperforms both homodyne and heterodyne detection whenever the channel exhibits “very strong” interference. We determine the capacity region for homodyne or heterodyne detection when the channel has “strong” interference, and we conjecture the existence of a joint detection strategy that outperforms the former two strategies in this case. Finally, we determine the Han-Kobayashi achievable rate regions for both homodyne and heterodyne detection and compare them to a region achievable by a conjectured joint detection strategy. In these latter cases, we determine achievable rate regions if the receivers employ a recently discovered minentropy quantum simultaneous decoder.