Transformations between symmetric sets of quantum states

Abstract

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.