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 Xi and Pi at each mode i=A,B,C. The first monogamy inequality involves the well-known quantity DIJ defined by Duan-Giedke-Cirac-Zoller as the sum of the variances of (XI-XJ)/2 and (PI+PJ)/2 where [XI,PJ]=δIJ. Entanglement between I and J is certified if DIJ < 1. A second monogamy inequality involves the more general entanglement certifier EntIJ defined as the normalized product of the variances of XI-gXJ and PI+gPJ, where g is a real constant. The monogamy inequalities give a lower bound on the values of DBC and EntBC for one pair, given the values DBA and EntBA 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 DAB=0.5 where there is symmetry between modes A and C.