  We calculate the quantum Renyi entropy in a phase space representation for
either fermions or bosons. This can also be used to calculate purity and
fidelity, or the entanglement between two systems. We show that it is possible
to calculate the entropy from sampled phase space distributions in normally
ordered representations, although this is not possible for all quantum states.
We give an example of the use of this method in an exactly soluble thermal
case. The quantum entropy cannot be calculated at all using sampling methods in
classical symmetric (Wigner) or antinormally ordered (Husimi) phase spaces, due
to inner product divergences. The preferred method is to use generalized
Gaussian phase space methods, which utilize a distribution over stochastic
Green's functions. We illustrate this approach by calculating the reduced
entropy and entanglement of bosonic or fermionic modes coupled to a
time-evolving, non-Markovian reservoir.
