Annales Henri Poincaré

, Volume 19, Issue 10, pp 2919–2953 | Cite as

The One-Mode Quantum-Limited Gaussian Attenuator and Amplifier Have GaussianMaximizers

  • Giacomo De PalmaEmail author
  • Dario Trevisan
  • Vittorio Giovannetti


We determine the \(p\rightarrow q\) norms of the Gaussian one-mode quantum-limited attenuator and amplifier and prove that they are achieved by Gaussian states, extending to noncommutative probability the seminal theorem “Gaussian kernels have only Gaussian maximizers” (Lieb in Invent Math 102(1):179–208, 1990). The quantum-limited attenuator and amplifier are the building blocks of quantum Gaussian channels, which play a key role in quantum communication theory since they model in the quantum regime the attenuation and the noise affecting any electromagnetic signal. Our result is crucial to prove the longstanding conjecture stating that Gaussian input states minimize the output entropy of one-mode phase-covariant quantum Gaussian channels for fixed input entropy. Our proof technique is based on a new noncommutative logarithmic Sobolev inequality, and it can be used to determine the \(p\rightarrow q\) norms of any quantum semigroup.


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Authors and Affiliations

  • Giacomo De Palma
    • 1
    • 2
    • 3
    Email author
  • Dario Trevisan
    • 4
  • Vittorio Giovannetti
    • 2
  1. 1.QMATH, Department of Mathematical SciencesUniversity of CopenhagenCopenhagenDenmark
  2. 2.NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNRPisaItaly
  3. 3.INFNPisaItaly
  4. 4.Università degli Studi di PisaPisaItaly

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