Bosonic Quantum Communication Across Arbitrarily High Loss Channels
Ludovico Lami, Martin B. Plenio, Vittorio Giovannetti, Alexander Semenovich Holevo
Abstract
A general attenuator ${\mathrm{\ensuremath{\Phi}}}_{\ensuremath{\lambda},\ensuremath{\sigma}}$ is a bosonic quantum channel that acts by combining the input with a fixed environment state $\ensuremath{\sigma}$ in a beam splitter of transmissivity $\ensuremath{\lambda}$. If $\ensuremath{\sigma}$ is a thermal state, the resulting channel is a thermal attenuator, whose quantum capacity vanishes for $\ensuremath{\lambda}\ensuremath{\le}1/2$. We study the quantum capacity of these objects for generic $\ensuremath{\sigma}$, proving a number of unexpected results. Most notably, we show that for any arbitrary value of $\ensuremath{\lambda}>0$ there exists a suitable single-mode state $\ensuremath{\sigma}(\ensuremath{\lambda})$ such that the quantum capacity of ${\mathrm{\ensuremath{\Phi}}}_{\ensuremath{\lambda},\ensuremath{\sigma}(\ensuremath{\lambda})}$ is larger than a universal constant $c>0$. Our result holds even when we fix an energy constraint at the input of the channel, and implies that quantum communication at a constant rate is possible even in the limit of arbitrarily low transmissivity, provided that the environment state is appropriately controlled. We also find examples of states $\ensuremath{\sigma}$ such that the quantum capacity of ${\mathrm{\ensuremath{\Phi}}}_{\ensuremath{\lambda},\ensuremath{\sigma}}$ is not monotonic in $\ensuremath{\lambda}$. These findings may have implications for the study of communication lines running across integrated optical circuits, of which general attenuators provide natural models.