Mixed state quantum reference frame resources

Situations where the operations of a noisy channel used for the transmission and retrieval of quantum states belong to a specific group of transformations give rise to resources beside entanglement that allow us to overcome the ensuing constraints, such as when shared reference frames (RF) associated with symmetry groups are lacking between the nodes of a quantum channel. So far, most work on this new kind of resource, dubbed "frameness", has been focused on pure state transformations even though almost all states and operations in the lab involve some degree of mixedness. Here we address the problem of quantifying the frameness of mixed states. We introduce a new family of pure state frameness measures associated with Abelian Lie groups in a Hilbert space of arbitrary but finite dimensions, whose convex roof extensions remain monotonic. In particular, we show that this family of frameness monotones are closely related to generalized concurrence functions of the reduced density operators of entangled states. This highlights interesting and deep links between frameness and entanglement resource theories, and provides a new way of classifying all frameness monotones as functions of the "twirled" state that results from tracing out the RF, where the state plus the RF are treated as a joint entangled system. Finally, we use a member of this family of frameness monotones to determine the explicit analytical form of a qubit's frameness of formation. The frameness of formation denotes the minimum average cost of preparing the ensemble of pure states that realize a given mixed state, and can be used to quantify the frameness of that state under certain conditions. Our results thus extends Wootter's formula for the entanglement of formation of bipartite qubit states to a whole new and different class of resources.