Monogamy inequalities for certifiers of continuous-variable Einstein-Podolsky-Rosen entanglement without the assumption of Gaussianity

We consider three modes $A$, $B$, and $C$ and derive monogamy inequalities that constrain the distribution of bipartite continuous variable Einstein-Podolsky-Rosen entanglement amongst the three modes. The inequalities hold without the assumption of Gaussian states, and are based on measurements of the quadrature phase amplitudes $X_i$ and $P_i$ at each mode $i=A,B,C$. The first monogamy inequality involves the well-known quantity $D_{IJ}$ defined by Duan-Giedke-Cirac-Zoller as the sum of the variances of $(X_I-X_J)/2$ and $(P_I+P_J)/2$ where $[X_I,P_J]={\delta }_{IJ}$. Entanglement between $I$ and $J$ is certified if $D_{IJ}<1$. A second monogamy inequality involves the more general entanglement certifier ${\text{Ent}}_{IJ}$ defined as the normalized product of the variances of $X_I-g{X}_J$ and $P_I+g{P}_J$, where $g$ is a real constant. The monogamy inequalities give a lower bound on the values of $D_{BC}$ and ${\text{Ent}}_{BC}$ for one pair, given the values $D_{BA}$ and ${\text{Ent}}_{BA}$ for the first pair. This lower bound changes in the absence of two-mode Gaussian steering of $B$. We illustrate for a range of tripartite entangled states, identifying regimes of saturation of the inequalities. The monogamy relations explain without the assumption of Gaussianity the experimentally observed saturation at $D_{AB}=0.5$ where there is symmetry between modes $A$ and $C$.

Figure 1

Entanglement monogamy: The entanglement is shared in a way that ensures $D_{BA}+D_{BC}\ge 1$. Here, $D_{AB}<1$ is a certifier of bipartite EPR entanglement between $A$ and $B$. Bipartite EPR entanglement is depicted by the dashed lines, and increases as $D_{IJ}\to 0$. The inequality Eq. (2) places bounds on the more general bipartite EPR entanglement certifiers ${\text{Ent}}_{AB}$ and ${\text{Ent}}_{BC}$, in terms of bipartite symmetry parameters $g_{BA}^{\left(\text{sym}\right)}$ and $g_{BC}^{\left(\text{sym}\right)}$. Stricter conditions apply if steering of system $B$ by the combined systems $AC$ cannot be certified using a standard steering criterion.

Figure 2

Configuration for the generation of a CV tripartite entangled system. A two-mode squeezed state is generated using a parametric down conversion (PDC) process. Two-mode entanglement can also be created using single-mode squeezed vacuum states that are input to a beam splitter $\text{BS}1$. Either way, entangled modes $B$ and $F$ are created at the outputs of the first device. Final modes $A$ and $C$ are then created at the output of the second beam splitter $\text{BS}2$, which has a transmission efficiency ${\eta }_0$ and a second vacuum input. CV $W$-type states are created with a coherent vacuum input; CV GHZ states are created with a squeezed vacuum input.

Figure 3

Monogamy for the CV $W$-type tripartite state. $D_{BA}$, $D_{BC}$, and $D_{BA}+D_{BC}$ vs ${\eta }_0$ for the state of Fig. 2 with a coherent vacuum input to $\text{BS}2$. Here $r=0.5$ (a) and $r=2$ (b). The monogamy bound of 1 is indicated by the gray dotted line.

Figure 4

Monogamy of the GMVT entanglement certifier $\text{Ent}$ for the CV $W$-type tripartite state. ${\text{Ent}}_{BA}$, ${\text{Ent}}_{BC}$, ${\text{Ent}}_{BA}{\text{Ent}}_{BC}$, and the monogamy bound $M_B$ (gray dotted line) are plotted vs ${\eta }_0$. The symmetry parameters $g_{BA}^{\left(\text{sym}\right)}\text{and}{g}_{BC}^{\left(\text{sym}\right)}$ are plotted in (c, d). Here $r=0.5$ (a, c) and $r=2$ (b, d). Saturation of the monogamy relation (27) is obtained for all ${\eta }_0$ for larger $r$ (b), where the gray dotted and green dashed lines coincide.

Figure 5

Monogamy for the CV $W$-type tripartite state of Fig. 2 with equal observed losses for modes $B$ and $A$ (${\eta }_B={\eta }_0$). $D_{BA}$, $D_{BC}$, and $D_{BA}+D_{BC}$ are plotted vs ${\eta }_B$ where mode $B$ has loss ${\eta }_B={\eta }_0$. Here $r=0.5$ (left) and $r=2$ (right). Plot (b) shows the saturation ($D_{BA}+D_{BC}=1$) of the monogamy relation (4) that occurs for large $r$ at ${\eta }_0=0.5$ in this case. The monogamy bound of 1 is indicated by the gray dotted line.

Figure 6

Monogamy for the CV $W$-type tripartite state with equal observed losses for modes $B$ and $A$ (${\eta }_B={\eta }_0$). ${\text{Ent}}_{BA}$, ${\text{Ent}}_{BC}$, ${\text{Ent}}_{BA}{\text{Ent}}_{BC}$, and the monogamy bound $M_B$ (gray dotted line) are plotted vs ${\eta }_B$. Here $r=0.5$ (a,c) and $r=2$ (b,d). Plots (c) and (d) are the symmetry parameters vs ${\eta }_B$. The symmetry parameters satisfy $g_{BA}^{\left(\text{sym}\right)}=1$ (for all ${\eta }_0$) and $g_{BC}^{\left(\text{sym}\right)}=1$ only for ${\eta }_0=0.5$.

Figure 7

Monogamy for the tripartite state of Fig. 2 with extra losses for mode $B$. ${\text{Ent}}_{BA}$, ${\text{Ent}}_{BC}$, ${\text{Ent}}_{BA}{\text{Ent}}_{BC}$, and the monogamy bound $M_B$ (gray dotted line) are plotted vs ${\eta }_B$ for $r=2$. The symmetry parameters are plotted in (c) and (d). Here we take ${\eta }_0=0.5$ (a,c) and ${\eta }_0=0.8$ (b,d). Saturation is observed for the symmetric case of plot (a) over all ${\eta }_B$ (the gray dotted and green dashed lines coincide).

Figure 8

Monogamy for the CV $W$-type state with extra losses for modes $A$ and $C$. $D_{BA}$, $D_{BC}$, and $D_{BA}+D_{BC}$ are plotted vs ${\eta }_A$ (${\eta }_B=1$, $r=2$). The monogamy bound is shown by the gray dotted line. Plots (a) and (b) assume ${\eta }_A={\eta }_C$ and ${\eta }_0=0.5$ (a) and ${\eta }_0=0.8$ (b). Plots (c) and (d) show values for ${\eta }_C=1$. Plot (e) gives the steering parameter $S_{B\left|\right\{AC\}}$ [Eq. (35)] in each case. $S_{B\left|\right\{AC\}}$ is independent of ${\eta }_0$.

Figure 9

Monogamy for the CV $W$-type tripartite state with extra losses for modes $A$ and $C$. ${\text{Ent}}_{BA}$, ${\text{Ent}}_{BC}$, and ${\text{Ent}}_{BA}{\text{Ent}}_{BC}$ are plotted vs ${\eta }_A$ where ${\eta }_B=1$ for $r=2$. The monogamy bound $M_B$ is shown by the gray dotted line. Plots (a) and (b) assume ${\eta }_A={\eta }_C$ where ${\eta }_0=0.5$ (a) and ${\eta }_0=0.8$ (b). Plots (c) and (d) show values where ${\eta }_C=1$. The steering parameter is given by Fig. 8. Plot (e) gives the symmetry parameter $g_{BA}^{\left(\mathrm{sym}\right)}$ and $g_{BC}^{\left(\mathrm{sym}\right)}$ as a function of ${\eta }_A$. For ${\eta }_A={\eta }_C$, $g_{BA}^{\left(\mathrm{sym}\right)}=g_{BC}^{\left(\mathrm{sym}\right)}$. When this is not the case, $g_{BA}^{\left(\mathrm{sym}\right)}$ and $g_{BC}^{\left(\mathrm{sym}\right)}$ vs ${\eta }_A$ are plotted, fixing ${\eta }_C=1$.