Gaussification and Entanglement Distillation of Continuous-Variable Systems: A Unifying Picture

Distillation of entanglement using only Gaussian operations is an important primitive in quantum communication, quantum repeater architectures, and distributed quantum computing. Existing distillation protocols for continuous degrees of freedom are only known to converge to a Gaussian state when measurements yield precisely the vacuum outcome. In sharp contrast, non-Gaussian states can be deterministically converted into Gaussian states while preserving their second moments, albeit by usually reducing their degree of entanglement. In this work—based on a novel instance of a noncommutative central limit theorem—we introduce a picture general enough to encompass the known protocols leading to Gaussian states, and new classes of protocols including multipartite distillation. This gives the experimental option of balancing the merits of success probability against entanglement produced.

Figure 1

(i) A single step of the general class of protocols (GG) considered, illustrated for three parties. This embodies the known Gaussifier (GP) entanglement distillation schemes based on projections onto the vacuum, or the extremality protocol (EP) mapping unknown states onto Gaussian ones with the same second moments. The Gaussian projection can be arbitrary [including (ii) vacuum projection, (iii) homodyning, (iv) eight-port homodyning, (v) tracing out], and schemes with finite widths of acceptance.

Figure 2

The degree of entanglement of the initial state $\rho$ in terms of the log-negativity versus the target Gaussian ${\rho }_{\infty }$ for (i) bipartite state vectors $|{\Psi }_{\lambda }⟩$ and (ii) tripartite state vectors $|{\Phi }_{\mu }⟩$. In both figures the variable $0<\Delta <1$ controls the degree of postselection. The curve varying with $\Delta$ is the entanglement of ${\rho }_{\infty }$, and the curve constant in $\Delta$ is the initial entanglement of $\rho$.