Continuous-variable entanglement distillation and noncommutative central limit theorems

Entanglement distillation transforms weakly entangled noisy states into highly entangled states, a primitive to be used in quantum repeater schemes and other protocols designed for quantum communication and key distribution. In this work, we present a comprehensive framework for continuous-variable entanglement distillation schemes that convert noisy non-Gaussian states into Gaussian ones in many iterations of the protocol. Instances of these protocols include (a) the recursive-Gaussifier protocol, (b) the temporally reordered recursive-Gaussifier protocol, and (c) the pumping-Gaussifier protocol. The flexibility of these protocols gives rise to several beneficial trade-offs related to success probabilities or memory requirements, which can be adjusted to reflect experimental demands. Despite these protocols involving measurements, we relate the convergence in this protocol to new instances of noncommutative central limit theorems, in a formalism that we lay out in great detail. Implications of the findings for quantum repeater schemes are discussed.

Figure 1

An implementation of an individual building block with beam-splitter reflectivity $R$ for (a) single-mode states and (b) two-mode states. Generalization to $m$-mode states is straightforward as each additional party performs the same local unitaries.

Figure 2

Different protocols combining building blocks in ways. All building blocks use the same filter, $\Pi$, and are labeled with their beam-splitter reflectivity $R$. (a) The recursive-Gaussifier protocol; (b) the temporally reordered recursive-Gaussifier protocol; (c) the pumping-Gaussifier protocol; and (d) the compact distillery protocol. The key shows how the building block labels compare with the variables used in Sec. 2.

Figure 3

A schematic of the CV repeater network considered here. Sources are assumed to produce a pure two-mode squeezed state of some chosen squeezing. Channels are predominately affected by attenuation, but also a small amount of room-temperature thermal noise. De-Gaussification is performed by photon replacement. Gaussification is performed as described here and in previous work using measurements projecting onto the vacuum, using many copies of the de-Gaussified state and asymptotically approaching a Gaussian state. Entanglement swapping uses deterministic optimal continuous swapping protocol.

Figure 4

The maximum attainable distance $L$ for a range of initial squeezing $r$ for which entanglement can be distributed. In plot (i) noise arises wholly from transmission between repeater stations, whereas in plot (ii) we incur an additional $50\%$ attenuation within each repeater station (see text for more details). Regions are shown for direct transmission and CV quantum repeater networks dividing the distance up into $m$ intervals. Some noteworthy terrestrial scales are shown at the top.