Continuous-variable entanglement distillation over a pure loss channel with multiple quantum scissors

Entanglement distillation is a key primitive for distributing high-quality entanglement between remote locations. Probabilistic noiseless linear amplification based on the quantum scissors is a candidate for entanglement distillation from noisy continuous-variable (CV) entangled states. Being a non-Gaussian operation, the quantum scissors is challenging to analyze. We present a derivation of the non-Gaussian state heralded by multiple quantum scissors in a pure loss channel with two-mode squeezed vacuum input. We choose the reverse coherent information (RCI), a proven lower bound on the distillable entanglement of a quantum state under one-way local operations and classical communication (LOCC), as our figure of merit. We evaluate a Gaussian lower bound on the RCI of the heralded state. We show that it can exceed the unlimited two-way LOCC-assisted direct transmission entanglement distillation capacity of the pure loss channel. The optimal heralded Gaussian RCI with two quantum scissors is found to be significantly more than that with a single quantum scissors, albeit at the cost of decreased success probability. Our results fortify the possibility of a quantum repeater scheme for CV quantum states using the quantum scissors.

Figure 1

(a) Noiseless linear amplification implemented using linear optics and (b) the quantum scissors operation based on photon injection and detection. The green machines in (a) refer to an $n$-mode input $n$-mode device that performs an $n$-way splitting or its inverse (recombining) unitary operation. When all but one of the input modes are in the vacuum state, the green machine splits the mean photon number in the first input mode uniformly across all the output modes.

Figure 2

Pure loss channel of transmissivity $\eta$ with TMSV state input appended by $N$-quantum scissors of gain $g=\sqrt{(1-\kappa )/\kappa }$ at the output.

Figure 3

Modified quantum scissors implemented with a weak TMSV source and on-off projections, appended to a pure loss channel of transmissivity $\eta$ with TMSV state input. When the idler mode of the weak TMSV state is projected on $I-|0\rangle \langle 0|$ a single photon is heralded in the signal mode and injected into the quantum scissors.

Figure 4

Gaussian RCI heralded across a pure loss channel of transmissivity $\eta =0.01$ appended with (a) $N=1$ NLA and (b) $N=2$ NLA as a function of the input TMSV state mean photon number and the gain of the NLA. The parameter $\kappa$ is related to the NLA gain by $\kappa =1/(1+g^2)$. The floors of the plots correspond to $C_{\mathrm{direct}}\left(\eta \right)$. The roughness in the surface in (b) is due to the finiteness of precision in our numerics.

Figure 5

Heralding success probability with $N=1,2$ NLA deployed on a pure loss channel of transmissivity $\eta =0.01$, as a function of the input TMSV state mean photon number and the gain of the NLA. The parameter $\kappa$ is related to the NLA gain by $\kappa =1/(1+g^2)$.

Figure 6

Gaussian RCI heralded across a pure loss channel of transmissivity $\eta =0.1$ appended with $N=1,2$ NLA, as a function of the input TMSV state mean photon number and the gain of the NLA. The parameter $\kappa$ is related to the gain of the NLA by $\kappa =1/(1+g^2)$. The planes in the plots correspond to $C_{\mathrm{direct}}\left(\eta \right)$.

Figure 7

Gaussian RCI heralded across $N=1,2$ NLA vs heralding success probability for a pure loss channel of transmissivity $\eta =0.01$.

Figure 8

One mode of a TMSV state teleported through a resource state consisting of a lossy TMSV state aided by an $N=1$ NLA.

Figure 9

Entanglement of formation of the covariance matrix heralded across the EC box of [34] for a TMSV input, as a function of the heralded effective transmission. The bare channel transmissivity is chosen to be $\eta =0.01$, the mean photon number of the input TMSV and the teleportation resource TMSV are chosen to be equal ${\mu }_{\mathrm{res}}=\mu =0.33$ and $N=1,2$, respectively, and the NLA gain $g$ is varied. (a) The EOF of the covariance matrix of the average state heralded across the error-corrected channel. (b) The average of EOF of the conditional heralded covariance matrices (conditioned on and averaged over the outcome of the dual homodyne detection). The black bold line corresponds to the EOF for transmission across the bare lossy channel.

Figure 10

Success probability of the EC box from [34] for $N=1,2$ NLA as a function of the heralded effective transmission. The bare channel transmissivity is chosen to be $\eta =0.01$, the mean photon number of the input TMSV and the teleportation resource TMSV are chosen to be equal ${\mu }_{\mathrm{res}}=\mu =0.33$ and $N=1,2$, respectively, and the NLA gain $g$ is varied.

Figure 11

The EOF and success probability of the covariance matrix heralded by the EC box for $N=1,2$ NLA as a function of the NLA intensity gain $g^2$.

Figure 12

Reverse coherent information of the covariance matrix heralded across the scheme in Fig. 8 with $N=1,2$ NLA and $\eta =0.01$. The parameter $\kappa$ is related to the gain of the NLA by $\kappa =1/(1+g^2)$.

Figure 13

Numerically optimized value of the product of heralded Gaussian RCI and the heralding success probability for $N=1,2$ NLA in the setup shown in Fig. 2, as a function of the pure loss channel's transmissivity $\eta$. The value is optimized over the NLA gain and the mean photon number of the input TMSV state. The black curve corresponds to the direct transmission capacity $C_{\mathrm{direct}}\left(\eta \right)=-{log}_2(1-\eta )$.