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arXiv:0706.3416 · Physical Review A

Classical capacity of bosonic broadcast communication and a minimum output entropy conjecture

Saikat Guha, Jeffrey H. Shapiro, Baris I. Erkmen

Published Sep 4, 2007• 75 citations
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Abstract

Previous work on the classical information capacities of bosonic channels has established the capacity of the single-user pure-loss channel, bounded the capacity of the single-user thermal-noise channel, and bounded the capacity region of the multiple-access channel. The latter is a multiple-user scenario in which several transmitters seek to simultaneously and independently communicate to a single receiver. We study the capacity region of the bosonic broadcast channel, in which a single transmitter seeks to simultaneously and independently communicate to two different receivers. It is known that the tightest available lower bound on the capacity of the single-user thermal-noise channel is that channel's capacity if, as conjectured, the minimum von Neumann entropy at the output of a bosonic channel with additive thermal noise occurs for coherent-state inputs. Evidence in support of this minimum output entropy conjecture has been accumulated, but a rigorous proof has not been obtained. We propose a minimum output entropy conjecture that, if proved to be correct, will establish that the capacity region of the bosonic broadcast channel equals the inner bound achieved using a coherent-state encoding and optimum detection. We provide some evidence that supports this conjecture, but again a full proof is not available.

DOI
10.1103/physreva.76.032303

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Previous work on the classical information capacities of bosonic channels has established the capacity of the single-user pure-loss channel, bounded the capacity of the single-user thermal-noise channel, and bounded the capacity region of the multiple-access channel. The latter is a multiple-user scenario in which several transmitters seek to simultaneously and independently communicate to a single receiver. We study the capacity region of the bosonic broadcast channel, in which a single transmitter seeks to simultaneously and independently communicate to two different receivers. It is known that the tightest available lower bound on the capacity of the single-user thermal-noise channel is that channel's capacity if, as conjectured, the minimum von Neumann entropy at the output of a bosonic channel with additive thermal noise occurs for coherent-state inputs. Evidence in support of this minimum output entropy conjecture has been accumulated, but a rigorous proof has not been obtained. We propose a minimum output entropy conjecture that, if proved to be correct, will establish that the capacity region of the bosonic broadcast channel equals the inner bound achieved using a coherent-state encoding and optimum detection. We provide some evidence that supports this conjecture, but again a full proof is not available.
We show that when coherent-state encoding is employed in conjunction with coherent detection, the Bosonic broadcast channel is equivalent to a classical degraded Gaussian broadcast channel whose capacity region is dual to that of the classical Gaussian multiple-access channel. We further show that if a minimum output-entropy conjecture holds true, then the ultimate classical information capacity of the Bosonic broadcast channel can be achieved by a coherent-state encoding. We provide some evidence in support of the conjecture.
Characterizing information-carrying capacities of bosonic communication channels is of significant practical interest. For thermal noise channels, using coherent (product) states yields an achievable rate for classical communication which is conjectured to be optimal. However, it is not known whether coding strategies using entanglement may perform better. Here we discuss upper bounds on classical capacities of thermal noise channels. These imply that coherent-state coding is close to optimal. Our main tool is a quantum analog of the entropy power inequality introduced Shannon. It gives a lower bound on the output von Neumann entropy when two independent signals combine at a beamsplitter.
We introduce a new form for the bosonic channel minimal output entropy conjecture, namely that among states with equal input entropy, the thermal states are the ones that have slightest increase in entropy when sent through a infinitesimal thermalizing channel. We then detail a strategy to prove the conjecture through variational techniques. This would lead to the calculation of the classical capacity of a communication channel subject to thermal noise. Our strategy detects input thermal ensembles as possible solutions for the optimal encoding of the channel, lending support to the conjecture. However, it does not seem to be able to exclude the possibility that other input ensembles can attain the channel capacity.
Determining the ultimate classical information carrying capacity of electromagnetic waves requires quantum-mechanical analysis to properly account for the bosonic nature of these waves. Recent work has established capacity theorems for bosonic single-user and broadcast channels, under the presumption of two minimum output entropy conjectures. Despite considerable accumulated evidence that supports the validity of these conjectures, they have yet to be proven. In this paper, it is shown that the second conjecture suffices to prove the classical capacity of the bosonic wiretap channel, which in turn would also prove the quantum capacity of the lossy bosonic channel. The preceding minimum output entropy conjectures are then shown to be simple consequences of an entropy photon-number inequality (EPnl), which is a conjectured quantum-mechanical analog of the entropy power inequality (EPI) from classical information theory.
We develop the capacity region for the bosonic broadcast channel in the presence of thermal noise and photon loss due to the environment. The bosonic channel is a quantum-mechanical description of many practical communication links such as optical, microwave, and radiofrequency. We employ our results to find the capacity region for quantum-secure covert broadcast over such channels and show that time-division is optimal as in the classical covert broadcast scenario. We rely on a strong minimum entropy output conjecture, a direct result of the entropy photon-number inequality (EPnI) conjecture.