On the proof of the majorization theorem for quantum Gaussian channels

Andrea Mari, Vittorio Giovannetti, Alexander Semenovich Holevo

Vittorio Giovannetti, Alexander Semenovich Holevo, Andrea Mari

Giacomo De Palma, Dario Trevisan, Vittorio Giovannetti

We prove that 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 maximizers (Lieb, Invent. Math. 102, 179 (1990)), stating that operators saturate the p->q norms of 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 channels.

Vittorio Giovannetti, A. S. Holevo, Raúl García−Patrón

Vittorio Giovannetti, Alexander Semenovich Holevo, Andrea Mari

We obtain a multimode extension of the majorization theorem for bosonic Gaussian channels, in particular, giving sufficient conditions under which the Glauber coherent states are the only minimizers for concave functionals of the output state of such a channel. We discuss direct implications of this multimode majorization for the positive solution of the famous additivity problem in the case of Gaussian channels.

Carlos Navarrete–Benlloch, Seth Lloyd, Jeffrey H. Shapiro +1 more

A long-standing open problem in quantum information theory is to find the classical capacity of an optical communication link, modeled as a Gaussian bosonic channel. It has been conjectured that this capacity is achieved by a random coding of coherent states using an isotropic Gaussian distribution in phase space. We show that proving a Gaussian minimum entropy conjecture for a quantum-limited amplifier is actually sufficient to confirm this capacity conjecture, and we provide a strong argument towards this proof by exploiting a connection between quantum entanglement and majorization theory.