Continuous-variable entanglement via multiphoton catalysis

We theoretically investigate the performance of multiphoton catalysis applied on the two-mode squeezed state by examining the entropy of entanglement, logarithmic negativity, Eistein-Podolsky-Rosen (EPR), and Hillery-Zubairy (HZ) correlations, and the fidelity of teleportation. It is found that the entanglement increases with the number of catalysis operations if the squeezing parameter is low initially. Our comparisons show that the HZ correlation presents a better performance than the EPR correlation for detecting the entanglement, and the improvement of HZ correlation definitely results in the improvement of entropy of entanglement rather than negativity; the region of enhanced EPR correlation is a subregion of all other entanglement properties. In addition, we consider the performances of the fidelity by comparing such operations applied before or after the amplitude damping channel. It is shown that the catalysis operation of $m=n=1$ before the channel presents the best performance in the initial-low squeezing regime. This may provide a useful insight for a long-distance quantum communication.

Figure 1

A single-mode optical state ${\left|\mathrm{\Phi }\right.〉}_{\mathrm{out}}$ is generated by the quantum catalysis. Here, an arbitrary input pure state ${\left|\psi \right.〉}_{\mathrm{in}}$ and an $m$-photon Fock state (at a beam splitter of reflectivity $R=1-T$), conditional measuring $m$ photons at one output.

Figure 2

An optical state $\left|\text{MC}\right.〉$ is generated by the quantum catalysis on both modes of the TMSV. $T_1={cos}^2\theta$ and $T_2={cos}^2\phi$ are the transmissivity of the beam splitters BS1 and BS2, respectively.

Figure 3

The success probability ($N_{m,n}^{-2}$) of detecting $m$ and $n$ photons on each mode as a function of the transmissivity $T$ and squeezing parameter $r$. (a) (b) and (c) (d) correspond to one- and two-mode catalyzed cases for $m=1,2$ with $T_1=T$ ($T_2=1$) and $m=n=1,2$ with $T_1=T_2=T$, respectively.

Figure 4

The difference of the log negativity between the catalyzed state and the TMSV as a function of the transmissivity $T$ and squeezing parameter $r$. The white region denotes negative values. (a)–(c) and (d)–(f) correspond to one- and two-mode catalyzed cases for $m=1,2,3$ with $T_1=T$ ($T_2=1$) and $m=n=1,2,3$ with $T_1=T_2=T$, respectively.

Figure 5

The logarithmic negativity is plotted as a function of (a) the transmissivity and (b) the squeezing parameter for several different values of $m$ and given $r=0.1$ and $T_1=0.1,$ respectively. Here for comparison, the corresponding logarithmic negativity of the TMSV is also plotted (see the dotted line).

Figure 6

The difference of entanglement between the catalyzed state and the TMSV (a) and (b) and their comparison (c)–(f) as a function of (a) and (b): $r,T_1$, and $T_2$; (c) and (d): $r,T_1=T_2=T$ with $m=n=1$; and (e) and (f): $r,T_1=T,T_2=1$ with $m=1$, 2, respectively. Entropy, dotted line, and negativity, dashed line, in (c)–(f).

Figure 7

The difference of EPR correlation between the catalyzed state and the TMSV as a function of (a) (b): $r,T_1$, and $T_2$; (c) (d): $r,T_1=T_2=T$ with $m=n=1,2$, respectively. Here for comparison, we plot the regions of enhanced entanglement compared to the TMSV (entropy, dotted line; negativity, dashed line).

Figure 8

The difference of HZ correlation ($l=1$) between the catalyzed state and the TMSV as a function of (a) (b): $r,T_1$ and $T_2$; (c) (d): $r,T_1=T_2=T$ with $m=n=1,2$, respectively. Here for comparison, we plot the regions of enhanced entanglement compared to the TMSV (entropy, dotted line; negativity, dashed line).

Figure 9

The HZ correlation of the catalyzed state as a function of $r$ with $T_1=T_2=T$ and $m=n=1,2$. Here for comparison, we plot the HZ correlation of the TMSV for $l=1,2$.

Figure 10

The difference of teleportation fidelity between the catalyzed state and the TMSV as a function of (a) (b): $r,T_1$ and $T_2$; (c) (d): $r,T_1=T_2=T$ with $m=n=1,2$, respectively. Here for comparison, we plot the regions of enhanced EPR (HZ) compared to the TMSV [dashed line (dotted line)].

Figure 11

The optimal fidelity as a function of the squeezing parameter for different values of $m=n$. Here for comparison, we plot the fidelity with the original TMSV in the dashed line (a) for coherent input, and (b) for squeezed vacuum input.

Figure 12

The optimal fidelity as a function of (a) the squeezing parameter for different values of $\mathcal{T}$, and (b) the photon-loss rate $\mathcal{T}$ for some given values of $r$. Here for comparison, we plot the fidelity with the original TMSV in the green-dashed line. For each case (associated with a special plot style), the corresponding lines are arranged from top to bottom with increasing (a) $\mathcal{T}=0.05,0.1,0.15$ and (b) $r=0.1,0.3,0.5,$ respectively. After catalysis, blue-dot lines; before catalysis, magenta dash-dot lines.