Transformations between symmetric sets of quantum states

We investigate probabilistic transformations of quantum states from a `source' set to a `target' set of states. Such transforms have many applications. They can be used for tasks which include state-dependent cloning or quantum state discrimination, and as interfaces between systems whose information encodings are not related by a unitary transform, such as continuous-variable systems and finite-dimensional systems. In a probabilistic transform, information may be lost or leaked, and we explain the concepts of leak and redundancy. Following this, we show how the analysis of probabilistic transforms significantly simplifies for symmetric source and target sets of states. In particular, we give a simple linear program which solves the task of finding optimal transforms, and a method of characterizing the introduced leak and redundancy in information-theoretic terms. Using the developed techniques, we analyse a class of transforms which convert coherent states with information encoded in their relative phase to symmetric qubit states. Each of these sets of states on their own appears in many well studied quantum information protocols. Finally, we suggest an asymptotic realization based on quantum scissors.