Characterizations of bilocality and $n$-locality of correlation tensors

In the literature, bilocality and $n$-locality of correlation tensors (CTs) are described by integration local hidden variable models (called C-LHVMs) rather than by summation LHVMs (called D-LHVMs). Obviously, C-LHVMs are easier to be constructed than D-LHVMs, while the later are easier to be used than the former, e.g., in discussing on the topological and geometric properties of the sets of all bilocal and of all $n$-local CTs. In this context, one may ask whether the two descriptions are equivalent. In the present work, we first establish some equivalent characterizations of bilocality of a tripartite CT $\mathbf{P}=\text{Lbrack}P(abc\mathrm{\mid }xyz)\text{Rbrack}$, implying that the two descriptions of bilocality are equivalent. As applications, we prove that all bilocal CTs with the same size form a compact path-connected set that has many star-convex subsets. Secondly, we introduce and discuss the bilocality of a tripartite probability tensor (PT) $\mathbf{P}=\text{Lbrack}P(abc)\text{Rbrack}$, including equivalent characterizations and properties of bilocal PTs. Lastly, we obtain corresponding results about $n$-locality of $n+1$-partite CTs $\mathbf{P}=\text{Lbrack}P(\mathbf{a}b\mathrm{\mid }\mathbf{x}y)\text{Rbrack}$ and PTs $\mathbf{P}=\text{Lbrack}P(\mathbf{a}b)\text{Rbrack}$, respectively.