Semiclassical and thermal phase space entropies measuring complexity

Abstract

Measures of delocalization in phase space are analyzed using Rényi entropies, especially two of which play an important role in characterizing extension and shape of distributions: the linear entropy related to the participation number and the Shannon-entropy. The difference of these two, termed as structural entropy, has been successfully applied in a large variety of physical situations and for various mathematical problems. A very similar quantity has coincidentally been used as a measure of complexity by some other authors. Hereby we show that various semiclassical phase space representations of quantum states can be well described by the structural entropy providing a transparent picture in relation to the thermodynamic description. Thermodynamic and quantum fluctuations are analytically treated for the special case of harmonic oscillators invoking the Einstein model of heat capacity. It is demonstrated that the thermal uncertainty relations are linked to the delocalization over the phase space. For respective limits of zero temperature implying quantum behavior or infinite temperature implying classical behavior we also show which quantities remain useful. As a byproduct the thermal extension of the phase space distribution can be calculated that is directly related to a decoherence parameter introduced by Zurek in a different context.