A Generalisation of Graph States to Two-Graph States
Constanza Riera
University of Bergen, Norway
Abstract: We present a class of states which contain the graph states. We show how to represent this larger class of states using a `two-graph’ description - whereas the graph state can be described by the expression (-1)p, for p a quadratic Boolean function, these more general states can be described by m(-1)p, where p is quadratic and m is a Boolean function formed from the product of affine terms. This polar form, m(-1)p, can be interpreted as a two-graph object, where p is a simple graph that describes the phases of the elements of the complex vector associated with the pure state, and m is a bipartite graph that describes the element magnitudes. This polar form yields an efficient, compact, and informative description of the orbit under certain unitaries from the local Clifford group of a graph state, via graph transforms on the two graphs. Conversely, the orbit of the two-graph state wrt the local Clifford group always contains a graph state.