The Metric of Bures and the Geometric Phase

Abstract

After the appearance of the papers of Berry [1], Simon [2], and of Wilczek and Zee [3], I tried to understand [4], whether there is a reasonable extension of the geometric phase - or, more accurately, of the accompanying phase factor - for general (mixed) states. A known recipe for such exercises is to use purifications: One looks for larger, possibly fictitious, quantum systems from which the original mixed states are seen as reductions of pure states. For density operators there is a standard way to do so by the use of Hilbert Schmidt operators (or by Hilbert Schmidt maps from an auxiliary Hilbert space into the original one).