Quantum states associated to mixed graphs and their algebraic characterization

Graph states are present in quantum information and found applications ranging from quantum network protocols (like secret sharing) to measurement based quantum computing. In this paper, we extend the notion of graph states, which can be regarded as pure quantum graph states, or as homogeneous quadratic Boolean functions associated to simple undirected graphs, to quantum states based on mixed graphs (graphs which allow both directed and undirected edges), obtaining mixed quantum states, which are defined by matrices associated to the measurement of homogeneous quadratic Boolean functions in some (ancillary) variables. In our main result, we describe the extended graph state as the sum of terms of a commutative subgroup of the stabilizer group of the corresponding mixed graph with the edges' directions reversed.