  In a quantum system, there may be many density matrices associated with a
state on an algebra of observables. For each density matrix, one can compute
its entropy. These are in general different. Therefore one reaches the
remarkable possibility that there may be many entropies for a given state
[Private communication from R. Sorkin]. This ambiguity in entropy can often be
traced to a gauge symmetry emergent from the non-trivial topological character
of the configuration space of the underlying system. It can also happen in
finite-dimensional matrix models. In the present work, we discuss this entropy
ambiguity and its consequences for an ethylene molecule. This is a very simple
and well-known system where these notions can be put to tests. Besides its
intrinsic conceptual interest, the simplicity of this model can serve as an
introduction to a similar discussion of systems such as coloured monopoles and
the breaking of colour symmetry.
