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AIP conference proceedings

Minimum Bosonic Channel Output Entropies

Vittorio Giovannetti

Published Jan 1, 2004• 6 citations
Publisher

Abstract

Coherent‐state inputs are conjectured to minimize the von Neumann entropies at the outputs of two Bosonic channels with thermal noise. Evidence in support of this conjecture is provided, including the fact that coherent‐state inputs minimize the integer‐order Rényi entropy and the Wehrl entropy at the outputs of these channels. A stronger conjecture—that output states resulting from coherent‐state inputs majorize the output states from other inputs—is also discussed.

DOI
10.1063/1.1834373

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Coherent‐state inputs are conjectured to minimize the von Neumann entropies at the outputs of two Bosonic channels with thermal noise. Evidence in support of this conjecture is provided, including the fact that coherent‐state inputs minimize the integer‐order Rényi entropy and the Wehrl entropy at the outputs of these channels. A stronger conjecture—that output states resulting from coherent‐state inputs majorize the output states from other inputs—is also discussed.
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.
We prove the long-standing conjecture stating that Gaussian thermal input states minimize the output von Neumann entropy of one-mode phase-covariant quantum Gaussian channels among all the input states with a given entropy. Phase-covariant quantum Gaussian channels model the attenuation and the noise that affect any electromagnetic signal in the quantum regime. Our result is crucial to prove the converse theorems for both the triple trade-off region and the capacity region for broadcast communication of the Gaussian quantum-limited amplifier. Our result extends to the quantum regime the entropy power inequality that plays a key role in classical information theory. Our proof exploits a completely new technique based on the recent determination of the p→q norms of the quantum-limited amplifier [De Palma et al., arXiv:1610.09967]. This technique can be applied to any quantum channel.