  Connections between the resource theories of coherence and purity (or
non-uniformity) are well known for discrete-variable, finite-dimensional,
quantum systems. We establish analogous results for continuous-variable
systems, in particular Gaussian systems. To this end, we define the concept of
maximal coherence at fixed energy, which is achievable with energy-preserving
unitaries. We show that the maximal Gaussian coherence (where states and
operations are required to be Gaussian) can be quantified analytically by the
relative entropy. We then propose a resource theory of non-uniformity, by
considering the purity of a quantum state at fixed energy as resource, and by
defining non-uniformity monotones. In the Gaussian case, we prove the equality
of Gaussian non-uniformity and maximal Gaussian coherence. Finally, we show a
hierarchy for non-uniformity, coherence, discord and entanglement in
continuous-variable systems.

Quantum Physics