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Journal of Physics A Mathematical and General

A lower bound on the information capacity of a quantum narrow-band free-space link

William G. Chambers

Published Oct 1, 1981• 0 citations
Publisher

Abstract

A method suggested by Holevo (1979) is used to obtain a lower bound for the information capacity of a quantum narrow-band free-space link without extraneous noise. At high photon rates the bound is better than those previously proposed and comes close to a fundamental upper bound. It is not so good at low rates, where it is beaten by photon-detecting systems. The normal modes are not treated as independent channels. A system is described where the normal modes are used in pairs.

DOI
10.1088/0305-4470/14/10/017

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A method suggested by Holevo (1979) is used to obtain a lower bound for the information capacity of a quantum narrow-band free-space link without extraneous noise. At high photon rates the bound is better than those previously proposed and comes close to a fundamental upper bound. It is not so good at low rates, where it is beaten by photon-detecting systems. The normal modes are not treated as independent channels. A system is described where the normal modes are used in pairs.
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.
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.
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 maximum rates for information transmission through noisy quantum channels has primarily been developed for memoryless channels, where the noise on each transmitted state is treated as independent. Many real-world communication channels experience noise which is modeled better by errors that are correlated between separate channel uses. In this Brief Report, upper bounds on the classical information capacities of a class of quantum memory channels are derived. The class of channels consists of indecomposable quantum memory channels, a generalization of classical indecomposable finite-state channels.
We present a family of easily computable upper bounds for the Holevo (information) quantity of an ensemble of quantum states depending on a reference state (as a free parameter). These upper bounds are obtained by combining probabilistic and metric characteristics of the ensemble. We show that an appropriate choice of the reference state gives tight upper bounds for the Holevo quantity which in many cases improve the estimates existing in the literature. We also present an upper bound for the Holevo quantity of a generalized ensemble of quantum states with finite average energy depending on the metric divergence of an ensemble. In the case of a multi-mode quantum oscillator, this upper bound is tight for large energy. Upper bounds for the Holevo capacity of finite-dimensional quantum channels depending on metric characteristics of the channel output are obtained.
We provide lower and upper bounds on the information transmission capacity of one single use of a classical-quantum channel. The lower bound is expressed in terms of the Hoeffding capacity, which we define similarly to the Holevo capacity but replacing the relative entropy with the Hoeffding distance. Similarly, our upper bound is in terms of a quantity obtained by replacing the relative entropy with the recently introduced max-relative entropy in the definition of the divergence radius of a channel.