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Journal of Physics A Mathematical and General

Quantum bounds on the information capacity of narrow-band free-space links without extraneous noise

William G. Chambers

Published Jan 1, 1981• 1 citation
Publisher

Abstract

An upper bound for the information rate in a communication link is set by the finite number of orthogonal states available to the electromagnetic field, is the link is subject to constraints on the average power and bandwidth. This bound may not be attainable in a free-space link since the transmitter does not have complete control over what is received. Some hypothetical systems are examined to see how close they get. At high photon rates it is possible with difficulty to improve slightly on an 'x-p' system, and at low photon rates on a photon-detecting system, but it seems that these systems are nearly the best, and certainly they are not far below the upper bound, in the sense that the ratios of the information rates to the maximum are close to unity. In all these systems the normal modes (Planck oscillators) are treated as independent channels. It is suggested that a beam splitter can provide a simple model for a free-space link.

DOI
10.1088/0305-4470/14/1/013

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An upper bound for the information rate in a communication link is set by the finite number of orthogonal states available to the electromagnetic field, is the link is subject to constraints on the average power and bandwidth. This bound may not be attainable in a free-space link since the transmitter does not have complete control over what is received. Some hypothetical systems are examined to see how close they get. At high photon rates it is possible with difficulty to improve slightly on an 'x-p' system, and at low photon rates on a photon-detecting system, but it seems that these systems are nearly the best, and certainly they are not far below the upper bound, in the sense that the ratios of the information rates to the maximum are close to unity. In all these systems the normal modes (Planck oscillators) are treated as independent channels. It is suggested that a beam splitter can provide a simple model for a free-space link.
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 information capacity of an optical channel under power constraints is ultimately limited by the quantum nature of transmitted signals. We discuss currently available and emerging photonic technologies whose combination can be shown theoretically to enable nearly quantum-limited operation of a noisy optical communication link in the photon-starved regime, with the information rate scaling linearly in the detected signal power. The key ingredients are quantum pulse gating to facilitate mode selectivity, photon-number-resolved direct detection, and a photon-efficient high-order modulation format such as pulse position modulation, frequency shift keying, or binary phase shift keyed Hadamard words decoded optically using structured receivers.
On the base of entropy defect principle the models are considered of narrow-band quantum communication channels and compared with the classic description of information channels. The classic model is shown to be applicable in a wide range of channels' parameters. But in the case of fermionic channels it is necessary to take into account that values of signal power and channel capacity are limited. These maximum values can be approximated by the simple expressions. Bosonic channels are free of these limitations. The rough quantum model of narrow-band bosonic channels gives too high values of capacity at low communication rate as compared with accurate one.
The capacity CCC of a communication channel is the maximum rate at which information can be transmitted without error from the channel's input to its output. The authors review quantum limits on the capacity that can be achieved with linear bosonic communication channels that have input power PPP. The limits arise ultimately from the Einstein relation that a field quantum at frequency fff has energy E=hfE=hfE=hf. A single linear bosonic channel corresponds to a single transverse mode of the bosonic field---i.e., to a particular spatial dependence in the plane orthogonal to the propagation direction and to a particular spin state or polarization. For a single channel the maximum communication rate is ${C}_{\mathrm{WB}}=(\frac{\ensuremath{\pi}}{\mathrm{ln}2})\sqrt{\frac{2P}{3h}}$ bits/s. This maximum rate can be achieved by a "number-state channel," in which information is encoded in the number of quanta in the bosonic field and in which this information is recovered at the output by counting quanta. Derivations of the optimum capacity CWB{C}_{\mathrm{WB}}CWB​ are reviewed. Until quite recently all derivations assumed, explicitly or implicitly, a number-state channel. They thus left open the possibility that other techniques for encoding information on the bosonic field, together with other ways of detecting the field at the output, might lead to a greater communication rate. The authors present their own general derivation of the single-channel capacity upper bound, which applies to any physically realizable technique for encoding information on the bosonic field and to any physically realizable detection scheme at the output. They also review the capacities of coherent communication channels that encode information in coherent states and in quadrature-squeezed states. A three-dimensional bosonic channel can employ many transverse modes as parallel single channels. An upper bound on the information flux that can be transferred down parallel bosonic channels is derived.
The information capacity of various communications systems is considered. Quantum effects are taken fully into account. The entropy of an electromagnetic wave having the quantum statistical properties of white noise in a single transmission mode is found, and from it the information efficiency of various possible systems may be derived. The receiving systems considered include amplifiers, heterodyne and homodyne converters and quantum counters. In the limit of high signal or noise power (compared to hνB, where h is Planck's constant and ν and B are, respectively, the center frequency and bandwidth of the channel) the information efficiency of an amplifier can approach unity. In the limit of low powers the amplifier becomes inefficient, while the efficiency of the quantum counter can approach unity. The amount of information that can be incorporated in a wave drops off rather rapidly when the power drops below hνB.