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

Abstract

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 $({X}_{I}\ensuremath{-}{X}_{J})/2$ and $({P}_{I}+{P}_{J})/2$ where $[{X}_{I},{P}_{J}]={\ensuremath{\delta}}_{IJ}$. 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 ${X}_{I}\ensuremath{-}g{X}_{J}$ 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 $C$.