Superselection rule-resource theory in the presence of partial prior knowledge

The study of new resource theories that arise from restrictions on possible quantum operations is turning into a very exciting field of research in quantum information theory. An interesting and remarkably fruitful approach has been to view the restrictions on quantum operations as coming from lack of access to classical reference frames. Superselection rules (SSR) are usually regarded to be axiomatic in nature. Surprisingly it turns out that the absence of reference frames can also give rise to superselection rules in a given situation. From a resource theory point of view this second outlook is much more productive. Each reference frame can be characterized by the group of its transformations. The corresponding superselection rule comes about by asking the allowed states and operations to remain invariant under the actions of this group. This is because such states and operations are all that can be prepared and implemented without the reference frame in question. Other states thus become resources in the presenceof the superselection rule [1]. Gour and Spekkens [2] have extensively studied the resources that arise from three types of superselection rules for pure unipartite quantum states: (1) Chirality or Z2-SSR, (2) phase reference or U(1)-SSR, and (3) (special cases of) Cartesian frames for spatial orientation or SU(2)-SSR. They have identified the form of allowed operations and the corresponding resources when no prior knowledge of the reference frame is assumed. They have found various relevant resource measures, the so called frameness measures, and developed their corresponding resource theories. A very interesting generalization of the above results is to consider the more practical case where the parties already have some knowledge of the reference frame, and the way this partial knowledge modifies the resource theory in question. This is introduced in the formalism by using a non-uniform measure overall possible transformations of the reference frame. Another challenging task is to broaden the scope ofthe theory to include mixed states. This is specially important since in real situations it is quite hard to prepare and work with pure states, and mixed states play a crucial role in almost all implementations of quantum information theory. Finally investigating the combined situation of mixed state resources in presence of prior partial knowledge and the interplay between the two can lead to many stimulating results. Our research shows that by restricting ourselves to pure states only in the cases studied, prior partial knowledge gives no new theory and leads to the identical set of resources as the case of completely unknown reference frame. To include mixed states, we extend the notion of frameness of formation in an analogous way to the entanglement of formation, as follows: The frameness measure of pure states in a given decomposition of the state in question are averaged with their relative weights. Minimizing this average over all possible decompositions then gives the frameness of formation for that mixed state. We have shown that a similar technique to Wootters’ regarding the entanglement of formation for bipartite states can be used to calculate the frameness of formation for a set of unipartite states in the Z2-SSR. This is a very interesting result since it shows that Wootters’ method also works in other resource theories, and is therefore more general than previously known. [References: [1] Reference frames, superselection rules, and quantum information, S. D. Bartlett, T. Rudolph,and R. W. Spekkens, Rev. Mod. Phys. 79, 555(2007) [2] The resource theory of quantum reference frames: manipulations and monotones, G. Gourand R. W. Spekkens, New Journal of Physics 10(2008) 033023 ]