Minimum output entropy of bosonic channels: A conjecture

By Vittorio Giovannetti, Saikat Guha, Seth Lloyd, Lorenzo Maccone, Jeffrey H. Shapiro. 2004

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The von Neumann entropy at the output of a bosonic channel with thermal noise is analyzed. Coherent-state inputs are conjectured to minimize this output entropy. Physical and mathematical evidence in support of the conjecture is provided. A stronger conjecture—that output states resulting from coherent-state inputs majorize the output states from other inputs—is also discussed. presented. We also consider a stronger conjecture—that out- put states resulting from coherent-state inputs majorize the output states from other inputs—which, if true, would imply the minimum output entropy conjecture. (Note that the mini- mum output entropy problem was previously treated in (7), which reported some of the results that we will discuss.) Additional supporting evidence for the minimum-entropy conjecture appears in our companion paper (8), where we show that the integer-order Renyi entropies and the Wehrl entropy at the output of the bosonic channel are minimized when the channel input is a coherent state. In Sec. II we present CP maps for the two bosonic chan- nels that will be considered in this paper. The minimum out- put entropy conjecture and its stronger (majorization) version are then stated and explained. In Sec. III we analyze the two channel maps in detail, and develop some useful properties of their output entropies. In Sec. IV we prove the minimum output entropy conjecture for the restricted scenario in which only Gaussian states may be fed into the channel. In Sec. V we present a collection of lower bounds on S. These lower bounds are consistent with the minimum output entropy con- jecture. Moreover, in the low- and high-noise regimes they approach asymptotically the upper bounds from which the conjecture arises. In Sec. VI we obtain necessary conditions on any input state that minimizes the output entropy. We demonstrate, in particular, that every coherent-state input produces an output state that achieves a local minimum of the output entropy. Finally, in Sec. VII we address the stron- ger version of the conjecture by exhibiting some evidence that output states produced by coherent-state inputs majorize all other output states. The paper is structured so that Secs. IV and VII may be read independently. The most technical parts of the derivations have been relegated to the Appendi- ces.

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/a971de6da2a92a4653288677120389ea14799022