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湘南工科大学紀要 = Memoirs of Shonan Institute of Technology

Capacity of Lossy Photon Channels with Input Constraints

Fumio Kanaya

Published Mar 25, 1996• 0 citations
OpenAlex

Abstract

Recently, considerable interest has emerged in quantum states of light fields other than a classical coherent state. One example of such nonclassical quantum states is the number state which features zero uncertainty for the photon number. Because of this unique property, the number states are promising as information bearing lights in direct-detection optical communications. However, if a channel is lossy, the benefit of using the number state is lessened because random photon deletions due to dissipative effect of the channel introduce inherent uncertainty to the photon number measurement at a receiver. Therefore, even in noiseless optical channels, the lossy effect must be taken into account as an important degradation factor which limits the channel performance. In this paper we compute the capacity of lossy photon channels which utilize the number state light under a certain input constraint. For binary and ternary PAM schemes, we obtain the capacity C (Γ) which is subject to the input constraint that the average number of photons per transmitted symbol can not exceed the fixed number Γ.

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Recently, considerable interest has emerged in quantum states of light fields other than a classical coherent state. One example of such nonclassical quantum states is the number state which features zero uncertainty for the photon number. Because of this unique property, the number states are promising as information bearing lights in direct-detection optical communications. However, if a channel is lossy, the benefit of using the number state is lessened because random photon deletions due to dissipative effect of the channel introduce inherent uncertainty to the photon number measurement at a receiver. Therefore, even in noiseless optical channels, the lossy effect must be taken into account as an important degradation factor which limits the channel performance. In this paper we compute the capacity of lossy photon channels which utilize the number state light under a certain input constraint. For binary and ternary PAM schemes, we obtain the capacity C (Γ) which is subject to the input constraint that the average number of photons per transmitted symbol can not exceed the fixed number Γ.
We calculate numerically the capacity of a lossy photon channel assuming photon number resolving detection at the output. We consider scenarios of input Fock and coherent states ensembles and show that the latter always exhibits worse performance than the former. We obtain capacity of a discrete-time Poisson channel as a limiting behavior of the Fock states ensemble capacity. We show also that in the regime of a moderate number of photons and low losses the Fock states ensemble with direct detection is beneficial with respect to capacity limits achievable with quadrature detection.
The effect of channel losses on the capacity of noiseless optical communication systems utilizing the number state is studied. Based on these findings, the photon efficiency of these systems is evaluated in the presence of losses. Then, by comparing the photon efficiency of the number-state pulse-position modulation (PPM) and the coherent-state PPM under certain realistic conditions, it is found that the former is always superior to the latter from the perspective of photon efficiency regardless of losses.>
We consider the effect of loss on quantum-optical communication channels. The channel based on direct detection of number states, which for a lossless transmission line would achieve the ultimate quantum channel capacity, is easily degraded by loss. The same holds true for the channel based on homodyne detection of squeezed states, which also is very fragile to loss. On the contrary, the ``classical'' channel based on heterodyne detection of coherent states is loss-invariant. We optimize the a priori probability for the squeezed-state and the number-state channels, taking the effect of loss into account. In the low power regime we achieve a sizeable improvement of the mutual information, and both the squeezed-state and the number-state channels overcome the capacity of the coherent-state channel. In particular, the squeezed-state channel beats the classical channel for total average number of photons N<8N<8N<8. However, for sufficiently high power the classical channel always performs as the best one. For the number-state channel we show that with a loss etalesssim.6\\eta\\lesssim .6etalesssim.6 the optimized a priori probability departs from the usual thermal-like behavior, and develops gaps of zero probability, with a considerable improvement of the mutual information (up to 70 % of improvement at low power for attenuation eta=.15\\eta=.15eta=.15).
Communication protocols based on quantum states of light have attracted increasing interest in the recent years, since they offer the possibility of enhancing either the communication capacity or the security by exploiting the very quantum nature of the information carriers. While first theoretical ideas and practical realizations started from discrete variables (DC) coding (practically based on the weak laser beams and SPDC-generated photon pairs), the continuous variables (CV) approach (supposing the use of coherent beams and entangled beam pairs) to the quantum information processing is aimed at the increase of the effectiveness and reliability of the quantum information protocols. Though being the subject to the environmental influence and thus performing always imperfectly, the CV-based quantum systems require the study of the effect the environment has on the quantum information processing schemes which is done by analyzing the perturbance of the corresponding quantum states. In this work we address M-ary communication channels based on photon-number continuous-variable multiphoton entangled states (PNES) in the conditions of the realistic lossy optical media.
The applications of the general formulae of channel capacity developed in the quantum information theory to evaluation of information transmission capacity of optical channel are interesting subjects. In this review paper, we will point out that the formulation based on only classical-quantum channel mapping model may be inadequate when one takes into account a power constraint for noisy channel. To define the power constraint well, we should explicitly consider how quantum states are conveyed through a transmission channel. Basing on such consideration, we calculate a capacity formula for an attenuated noisy optical channel with genreral Gaussian state input; this gives certain progress beyond the example in our former paper.
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.