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Quantum Entropic Ambiguities: Ethylene

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.