To define a minimal mathematical framework for isolating some of the characteristic properties of quantum entanglement, we introduce a generalization of the tensor product of graphs. Inspired by the notion of a density matrix, the generalization is a simple one: every graph can be obtained by addition modulo two, possibly with many summands, of tensor products of adjacency matrices. In light of this, we are still able to prove a combinatorial analogue of the Peres–Horodecki criterion for testing separability.
The pure states that can be uniquely determined among all (UDA) states by their marginals are essential to efficient quantum state tomography. We generalize the UDA states from the context of pure states to that of arbitrary (whether pure or mixed) states, motivated by the efficient state tomography of low-rank states. The concept of additivity of
k-UDA states for three different composite types of tensor product applies if the composite state of two
k-UDA states is still uniquely determined by the
k-partite marginals for the corresponding type of tensor product. We show that the additivity holds if one of the two initial states is pure and present the conditions under which the additivity holds for two mixed UDA states. One of the three composite types of tensor product is also adopted to construct genuinely multipartite entangled (GME) states. Therefore, it is effective to construct multipartite
k-UDA states with genuine entanglement by uniting the additivity of
k-UDA states and the construction of GME states.