Quantum Capacities of Bosonic Channels

We investigate the capacity of bosonic quantum channels for the transmission of quantum information. We calculate the quantum capacity for a class of Gaussian channels, including channels describing optical fibers with photon losses, by proving that Gaussian encodings are optimal. For arbitrary channels we show that achievable rates can be determined from few measurable parameters by proving that every channel can asymptotically simulate a Gaussian channel which is characterized by second moments of the initial channel. Along the way we provide a complete characterization of degradable Gaussian channels and those arising from teleportation protocols.

Figure 1

In order to obtain a Gaussian channel from an arbitrary quantum channel $T$ Bob (the sender) prepares $n$ instances of an entangled state $\psi$ half of which he sends through $T^{\otimes n}$. After applying two arrays of $50:50$ beam splitters to the output ${\rho }_T^{\otimes n}=[(T\otimes \mathrm{id})(\psi ){]}^{\otimes n}$ the $n$ reduced states will converge to a Gaussian state $\mathcal{G}({\rho }_T)$ (with the same CM as ${\rho }_T$) which can in turn be used to establish a Gaussian teleportation channel $T_G$.

Figure 2

Quantum capacity of a channel with photon losses as a function of the transmission length $l$ in terms of the absorption length $l_a$, i.e., $\eta =e^{-l/l_a}$. For quantum memories $l$ and $l_a$ are storage and decay time. The capacity vanishes for $l/l_a=\ln 2\approx 0.693$, where the channel can be considered to be part of a symmetric approximate cloning channel.