Calculus of Variations and Geometric Measure Theory

Giacomo De Palma - D. Trevisan - V. Giovannetti

One-mode quantum-limited Gaussian channels have Gaussian maximizers

created by trevisan on 10 Jun 2017



Inserted: 10 jun 2017

Year: 2016

ArXiv: 1610.09967 PDF


We prove that Gaussian states saturate the p->q norms of the one-mode quantum-limited attenuator and amplifier. The proof starts from the majorization result of De Palma et al., IEEE Trans. Inf. Theory 62, 2895 (2016), and is based on a new logarithmic Sobolev inequality. Our result extends to noncommutative probability the seminal theorem "Gaussian kernels have only Gaussian maximizers" (Lieb, Invent. Math. 102, 179 (1990)), stating that Gaussian operators saturate the p->q norms of Gaussian integral kernels. Our result also implies that the p->q norms of the thinning are saturated by geometric probability distributions. Moreover, the multimode extension of our result would imply the multiplicativity of the p->q norms of quantum-limited Gaussian channels.