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arXiv:quant-ph/0404037 · Physical Review A

Minimum Rényi and Wehrl entropies at the output of bosonic channels

Vittorio Giovannetti, Seth Lloyd, Lorenzo Maccone, Jeffrey H. Shapiro, Brent J. Yen

Published Aug 30, 2004• 58 citations
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Abstract

The minimum R\'enyi and Wehrl output entropies are found for bosonic channels in which the signal photons are either randomly displaced by a Gaussian distribution (classical-noise channel), or coupled to a thermal environment through lossy propagation (thermal-noise channel). It is shown that the R\'enyi output entropies of integer orders $z\ensuremath{\geqslant}2$ and the Wehrl output entropy are minimized when the channel input is a coherent state.

DOI
10.1103/physreva.70.022328

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The minimum R'enyi and Wehrl output entropies are found for bosonic channels in which the signal photons are either randomly displaced by a Gaussian distribution (classical-noise channel), or coupled to a thermal environment through lossy propagation (thermal-noise channel). It is shown that the R'enyi output entropies of integer orders z\ensuremath⩾2z\ensuremath{\geqslant}2z\ensuremath⩾2 and the Wehrl output entropy are minimized when the channel input is a coherent state.
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 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.