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arXiv:2012.02801 · IEEE Transactions on Communications

Capacity of a Lossy Photon Channel With Direct Detection

Karol Łukanowski, Marcin Jarzyna

Published Apr 27, 2021• 5 citations
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Abstract

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.

DOI
10.1109/tcomm.2021.3075714

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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.
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.
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 study the general coherent-state hypothesis testing problem and the capacity of the pure-loss optical channel with a coherent processing receiver (a receiver that uses coherent feedback control and direct detection). We describe the binary hypothesis minimum probability of error receiver as optimizing the communication efficiency at each instant, based on recursively updated knowledge of the receiver. Using this viewpoint, we give a natural generalization of the designs to general M-ary hypothesis testing problems. We analyze the information capacity with coherent receivers, and compare the result with that with direct detection receivers and with arbitrary quantum receivers (the Holevo limit), using the appropriate scalings in the low photon number regime.
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).
We study the problem of designing optical receivers to discriminate between multiple coherent states using coherent processing receivers---i.e., one that uses arbitrary coherent feedback control and quantum-noise-limited direct detection---which was shown by Dolinar to achieve the minimum error probability in discriminating any two coherent states. We first derive and reinterpret Dolinar's binary-hypothesis minimum-probability-of-error receiver as the one that optimizes the information efficiency at each time instant, based on recursive Bayesian updates within the receiver. Using this viewpoint, we propose a natural generalization of Dolinar's receiver design to discriminate MMM coherent states, each of which could now be a codeword, i.e., a sequence of NNN coherent states, each drawn from a modulation alphabet. We analyze the channel capacity of the pure-loss optical channel with a general coherent-processing receiver in the low-photon number regime and compare it with the capacity achievable with direct detection and the Holevo limit (achieving the latter would require a quantum joint-detection receiver). We show compelling evidence that despite the optimal performance of Dolinar's receiver for the binary coherent-state hypothesis test (either in error probability or mutual information), the asymptotic communication rate achievable by such a coherent-processing receiver is only as good as direct detection. This suggests that in the infinitely long codeword limit, all potential benefits of coherent processing at the receiver can be obtained by designing a good code and direct detection, with no feedback within the receiver.
We consider a photonic communication system wherein the photon detector suffers a random ‘dead time’ following each successful photon detection. If subsequent photon arrivals occur during the dead time, the information contained in the photons is assumed to be erased. We refer to such channels as photonic erasure channels and derive fundamental limits on the rate at which classical information can be transmitted on such channels. We assume photon arrivals according to a Poisson process, and consider two classes of detectors - paralyzable and nonparalyzable. We derive explicit expressions for the capacity of photonic erasure channels, for any general distribution of the dead times of the detector. For a photonic erasure channel with a nonparalyzable detector, we show that the capacity depends only on the expected dead time. On the other hand, with a paralyzable detector, the channel capacity depends on the dead time distribution through its Laplace transform.