Role of correlations in the two-body-marginal problem

Quantum properties of correlations have a key role in disparate fields of physics including quantum information processing, quantum foundations, and strongly correlated systems. We tackle a specific aspect of the fundamental quantum marginal problem: We address the issue of deducing the global properties of correlations of tripartite quantum states based on the knowledge of their bipartite reductions, focusing on relating specific properties of bipartite correlations to global correlation properties. We prove that strictly classical bipartite correlations may still require global entanglement and that unentangled (albeit not strictly classical) reductions may require global genuine multipartite entanglement rather than simple entanglement. On the other hand, for three qubits, the strict classicality of the bipartite reductions rules out the need for genuine multipartite entanglement. Our work sheds light on the relation between local and global properties of quantum states and on the interplay between classical and quantum properties of correlations.

Figure 1

Hierarchy of correlation classes and some possible compatibility sets. The set ${\mathcal{S}}_{\mathrm{FC}}$ is denoted by the line. The set ${\mathcal{S}}_{\mathrm{GME}}$ of genuinely multipartite states is the complement of ${\mathcal{S}}_{\mathrm{BS}}$ in the space of all states ${\mathcal{S}}_{\mathrm{all}}$. The reductions corresponding to the compatibility set ${\mathcal{C}}_1$ are only compatible with genuine multipartite entanglement. The reductions of the states in compatibility set ${\mathcal{C}}_2$ are compatible with entangled biseparable states and genuinely multipartite entangled states, but not separable states. The reductions defining ${\mathcal{C}}_3$ are compatible with a fully separable state as well as with entangled states, but not fully classical states.