Frameness of formation for a qubit

Almost all states and operations in the lab involve some degree of mixedness, so it is necessary to extend the results of the newly developed reference frame resource theories to include mixed states. We produce, for the first time, explicit results for a qubit's frameness of formation. The frameness of formation denotes the average resource cost of generating a mixed state. This cost is measured in terms of standard resource states, called refbits, that are chosen as units of frameness. In order to determine the exact value of this frameness measure, we develop a novel technique that generalizes Wootter's idea for entanglement of formation to a wide class of reference frame resource theories. We introduce the "concurrence of frameness" as a generalization of the concurrence measure to the case of reference frames. The concurrence of a resource state is explicitly determined, and the cost of preparing a resource is expressed as a simple function of this concurrence. This approach is applicable to resource measures of any given group of transformations associated with a superselection rule, as long as the related resource cost can be written as an explicit function of the concurrence of frameness. Finally, we demonstrate the application of our result to the resource theories of the groups Z_2 and U(1) that are associated with chiral and phase reference frames respectively.