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Quantum Optics Journal of the European Optical Society Part B

Realistic performance of the maximum information channel

Michael J. W. Hall, Michael J. E. O’Rourke

Published Jun 1, 1993• 31 citations
Publisher

Abstract

For a fixed mean signal energy, the optimal narrow-band bosonic channel is based on photodetection of a Bose-Einstein distribution of number states. A simple formula is obtained for the average error probability per symbol of this channel when degraded by Gaussian noise and inefficient detection. A general information bound is derived for narrow-band channels degraded by Gaussian noise, which yields a lower bound for the average probability of error per bit. Results are applicable to all number-state channels in the limit of low noise and high efficiency.

DOI
10.1088/0954-8998/5/3/005

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For a fixed mean signal energy, the optimal narrow-band bosonic channel is based on photodetection of a Bose-Einstein distribution of number states. A simple formula is obtained for the average error probability per symbol of this channel when degraded by Gaussian noise and inefficient detection. A general information bound is derived for narrow-band channels degraded by Gaussian noise, which yields a lower bound for the average probability of error per bit. Results are applicable to all number-state channels in the limit of low noise and high efficiency.
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 description of Gaussian noise for single-mode quantum fields is briefly reviewed and applied to calculate the maximal information properties of three quantum communication channels degraded by Gaussian noise. These channels are based on (i) heterodyne detection of coherent states, (ii) homodyne detection of squeezed states, and (iii) photodetection of number states. It is found that each channel can outperform the others for a given noise level, where the optimum channel is essentially determined by the product of the average noise and signal energies.
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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.
We demonstrate the achievability of a square root limit on the amount of information transmitted reliably and with low probability of detection (LPD) over the single-mode lossy bosonic channel if either the eavesdropper's measurements or the channel itself is subject to the slightest amount of excess noise. Specifically, Alice can transmit O(n)\mathcal{O}(\sqrt{n})O(n​) bits to Bob over nnn channel uses such that Bob's average codeword error probability is upper-bounded by an arbitrarily small δ>0δ>0δ>0 while a passive eavesdropper, Warden Willie, who is assumed to be able to collect all the transmitted photons that do not reach Bob, has an average probability of detection error that is lower-bounded by 1/2−ε1/2-ε1/2−ε for an arbitrarily small ε>0ε>0ε>0. We analyze the thermal noise and pure loss channels. The square root law holds for the thermal noise channel even if Willie employs a quantum-optimal measurement, while Bob is equipped with a standard coherent detection receiver. We also show that LPD communication is not possible on the pure loss channel. However, this result assumes Willie to possess an ideal receiver that is not subject to excess noise. If Willie is restricted to a practical receiver with a non-zero dark current, the square root law is achievable on the pure loss channel.
We find the maximum rate achievable in the private communication over a bosonic quantum channel with a fully Gaussian protocol based on optimal single-mode Gaussian measurements. This rate establishes a lower bound on the secret rate capacity of the channel. We focus on the class of phase-insensitive Gaussian channels. For the thermal-loss and thermal amplification channels, our results demonstrate the optimality, within the constraints of our analysis, of previously proposed protocols, while also providing a significantly simplified formula for their performance evaluation. For the added noise channel, our rate provides a better lower bound than any previously known.