Characterizing genuine multipartite quantum correlations in quantum physical systems has historically been a challenging problem in quantum information theory. More recently, however, the total correlation or multipartite information measure has been helpful in accomplishing this goal, especially with the multipartite symmetric quantum (MSQ) discord (Piani et al. 2008 Phys. Rev. Lett. 100, 090502. ( doi:10.1103/PhysRevLett.100.090502 )) and the conditional entanglement of multipartite information (CEMI) (Yang et al. 2008 Phys. Rev. Lett. 101, 140501. ( doi:10.1103/PhysRevLett.101.140501 )). Here, we apply a recent and significant improvement of strong subadditivity of quantum entropy (Fawzi & Renner 2014 ( http://arxiv.org/abs/1410.0664 )) in order to develop these quantities further. In particular, we prove that the MSQ discord is nearly equal to zero if and only if the multipartite state for which it is evaluated is approximately locally recoverable after performing measurements on each of its systems. Furthermore, we prove that the CEMI is a faithful entanglement measure, i.e. it vanishes if and only if the multipartite state for which it is evaluated is a fully separable state. Along the way, we provide an operational interpretation of the MSQ discord in terms of the partial state distribution protocol, which in turn, as a special case, gives an interpretation for the original discord quantity. Finally, we prove an inequality that could potentially improve upon the Fawzi–Renner inequality in the multipartite context, but it remains an open question to determine whether this is so.
For multipartite states, we consider a notion of
D symmetry. For a system of
N qubits, it coincides with the usual permutational symmetry. In the case of
N qudits ($d\ensuremath{\ge}3$), the
D symmetry is stronger than the permutational one. For the space of all
D-symmetric vectors in ${({\mathbb{C}}^{d})}^{\ensuremath{\bigotimes}N}$, we define a basis composed of vectors ${|{R}_{N,d;k}\ensuremath{\rangle}:\phantom{\rule{0.16em}{0ex}}0\ensuremath{\le}k\ensuremath{\le}N(d\ensuremath{-}1)}$ which are analogs of Dicke states. The aim of this paper is to discuss the problem of separability of
D-symmetric states which are diagonal in the basis ${|{R}_{N,d;k}\ensuremath{\rangle}}$. We show that if
N is even and $d\ensuremath{\ge}2$ is arbitrary then a positive partial transposition property is a necessary and sufficient condition of separability for
D-invariant diagonal states. In this way, we generalize results obtained by Yu [Phys. Rev. A 94, 060101(R) (2016)] and Wolfe and Yelin [Phys. Rev. Lett. 112, 140402 (2014)]. Our strategy is to use some classical mathematical results on a moment problem.